Vendor | : | HP |

Exam Code | : | HP0-460 |

Exam Name | : | Implementing HP XP12000/10000 Solution Fundamentals |

Questions and Answers | : | 75 Q & A |

Updated On | : | February 15, 2019 |

PDF Download Mirror | : | Pass4sure HP0-460 Dump |

Get Full Version | : | Pass4sure HP0-460 Full Version |

HP0-460 exam Dumps Source : Implementing HP XP12000/10000 Solution Fundamentals

Test Code : HP0-460

Test Name : Implementing HP XP12000/10000 Solution Fundamentals

Vendor Name : HP

Q&A : 75 Real Questions

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Autodesk, an award-winning American software company, has collaborated with HP and GE Additive to advance generative design application tools that work directly with 3D printers.

in accordance with Autodesk’s Netfabb and Fusion 360, the design-to-print workflow for additive manufacturing could be integrated in HP Multi Jet Fusion and GE Direct steel Laser Melting (DMLM) systems to enhance fast prototyping for construction-in a position 3D printed materials.

Robert Yancey, director of producing business approach at Autodesk, defined to Design news:

“To release the total price of HP MJF printers, you want a great design, a great cloth, and a superb print technique. Autodesk develops the design equipment and know-how, HP develops the print manner, and HP with their material partners develops the materials. All features are required to achieve optimum effectivity.”

The HP Jet Fusion 3D 4210/4200 Printing solution. image by the use of HP.A generative workflow

all through Formnext, HP introduced a partnership with Autodesk to allow generative design capabilities across the whole HP Multi Jet Fusion portfolio. Following this collaboration, Christoph Schell, President of 3D Printing and Digital Manufacturing, HP Inc., pointed out:

“Many industries akin to car, which goes via its greatest transformation in additional than a hundred years, want to new applied sciences and strategic partners like HP to aid them enhanced compete during this time of trade.”

“we're working with innovators to change the style they design and manufacture, unlocking new purposes, more manufacturing flexibility, and more suitable innovation, efficiency, and sustainability across their product construction lifecycle.”

in a similar way, Autodesk is working with GE Additive to streamline metallic additive manufacturing. the usage of GE Additive’s software algorithms, interfaces, and really expert facts models, this workflow will offer predictive insights for charge and timeline projections in the early ranges of design.

“Working with Autodesk will give a powerful design-to-print atmosphere for our customers, helping lower the limitations of additive adoption while accelerating a consumer’s time to first decent part,” spoke of Lars Bruns, govt utility chief at GE Additive.

3D printed motorbike elements from HP. image via HP/ Motus.enforcing an conclusion-to-conclusion workflow

As a mutual client of Autodesk and HP, Penumbra Engineering, a united statesgenerative design enterprise, these days used Autodesk’s design-to-print workflow to provide an ultrasonic sensing machine. This 3D printed half become designed to be lightweight and durable inside severe environments.

“The Penumbra case look at uncovers the price of HP working with Autodesk generative design expertise,” brought Yancey.

“We’re helping HP printers in Fusion 360 and Netfabb so that HP multi-jet fusion clients have the design and print prep tools they want. We’re working with HP to deliver guide for the new metal printers.”

A 3D printed handheld transducer created by Penumbra Engineering. image by the use of Penumbra Engineering,publish your nominations now for the 3D Printing industry Awards 2019.

also, for the newest 3D Printing trade updates subscribe to our publication, observe us on Twitter and like us on facebook.

in search of a sparkling start in the new year? seek advice from 3D Printing Jobs to commence your career in additive manufacturing.

Featured photograph indicates 3D printed components from HP. photograph via HP.

RESTON, VA--(Marketwired - Mar 27, 2014) - Carahsoft technology Corp., the relied on government IT solutions company, today introduced that it has been recognized with an HP PartnerOne Award for boom Reseller of the 12 months on the 2014 HP world accomplice conference.

The HP PartnerOne growth Reseller of the 12 months Americas, HP utility community award honors Carahsoft for their powerful efficiency during the last 12 months and demonstrating boom via innovation, powerful teamwork and an standard commitment to excellence.

"Carahsoft is honored to be recognized again by using HP for our aid of the company's public sector channel company and income efforts," mentioned Patrick Gallagher, vice chairman of HP solutions at Carahsoft. "The HP PartnerOne application has been a key to our success, featuring easy entry to enablement substances and enabling our team and our partners to grow their ability sets and optimize our clients' know-how investments."

"Congratulations to this 12 months's HP PartnerOne growth Award winners," pointed out Robert Makheja, Vice-President, Americas HP Federal utility revenue. "With a laser focus on proposing trade-main know-how solutions and features to our mutual customers, each and every of these partners has exemplified magnificent increase via innovation and a real dedication to partnership."

The HP PartnerOne Award winners were selected by a panel of HP Channel executives and offered to astounding HP companions who have established brilliant business efficiency, performed gigantic ordinary increase, and have delivered ingenious solutions to clients.

additional info on the 2014 HP world partner convention is attainable here.

About Carahsoft Carahsoft know-how agency is the relied on govt IT options provider. As a proper-ranked GSA time table Contract holder, Carahsoft serves as the grasp govt aggregator for many of its most fulfilling-of-breed companies, aiding an intensive ecosystem of manufacturers, resellers and consulting partners committed to assisting govt groups select and enforce the most suitable answer at the very best value.

The business's dedicated solutions Divisions proactively market, promote and convey HP, VMware, Adobe, Symantec, EMC, F5 Networks, crimson Hat, SAP, and Intelligence and creative products and capabilities amongst others. Carahsoft is consistently recognized by using its companions as a accurate revenue producer, and is listed yearly among the many industry's quickest-turning out to be businesses with the aid of CRN, Inc., Washington technology, The Washington publish, Washington enterprise Journal, and SmartCEO. consult with us at www.carahsoft.com.

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SANTA CLARA, Calif., Feb 07, 2019 (GLOBE NEWSWIRE via COMTEX) -- SANTA CLARA, Calif., Feb. 07, 2019 (GLOBE NEWSWIRE) -- Telenav [(R)] , Inc. TNAV, +7.83% a leading provider of connected car and location-based services, today released its financial results for the quarter ended Dec. 31, 2018, the second quarter of fiscal 2019. In connection with that announcement, the company also posted a quarterly letter to stockholders on its website. Please visit Telenav's investor relations website at http://investor.telenav.com to view the financial results and letter to stockholders.

"In the second quarter, we achieved a significant milestone of positive adjusted cash flow from operations, a non-GAAP measure, evidence that the company is improving its financial fundamentals while driving growth," said HP Jin, Chairman and CEO of Telenav. "We continued to increase our revenue and billings from General Motors, which were approximately 17% and 20% respectively during the second quarter compared to 7% and 7% for revenue and billings in the second quarter of fiscal 2018. Consistent with our mission of making people's lives less stressful, more productive, and more fun on the go, we are collaborating with Amazon to integrate the Alexa voice interface for in-car navigation, an integration we demonstrated at CES 2019."

Telenav's Board of Directors has authorized a program for the repurchase of up to $20.0 million of shares of common stock through open market purchases. The term of the program is 18 months. The timing and amount of repurchase transactions under this program will depend on market conditions, cash flow and other considerations.

Financial highlights for the second quarter ended Dec. 31, 2018

Recent Business Highlights

Q3 Fiscal 2019 Business Outlook

For the third fiscal quarter ending Mar. 31, 2019, Telenav offers the following guidance.

Subject to anticipated volumes, take rates and timing of model expansion under Telenav's various automobile manufacturer and tier one supplier programs, including the potential impact, if any, of our automotive manufacturer customers' transition of their North American passenger car portfolio to trucks, SUVs and CUVs, and assuming no unforeseen impact from macroeconomic changes, including federal government shutdowns and tariff impacts, Telenav anticipates that adjusted cash flow from operations will be positive for fiscal 2019.

The above information concerning guidance represents Telenav's outlook only as of the date hereof and is subject to change as a result of amendments to material contracts, other changes in business conditions and other factors. Please refer to the disclosures under "Forward-Looking Statements" below. Telenav undertakes no obligation to update or revise any financial forecast or other forward-looking statements, as a result of new developments, or otherwise.

Conference Call and Quarterly Commentary

Telenav will host an investor conference call and live webcast on Thursday, Feb. 7, 2019 at 2:30 p.m. Pacific Time (5:30 p.m. Eastern Time). Management has posted its letter to stockholders in combination with this press release on its investor relations website in lieu of management providing remarks at the start of the conference call. Instead, management will respond to questions during the call. To listen to the webcast and view Telenav's quarterly commentary, please visit Telenav's investor relations website at http://investor.telenav.com. Listeners can also access the conference call by dialing 888-394-8218 (toll-free, domestic only) or 323-794-2588 (domestic and international toll) and entering pass code 2400241. A replay of the conference call will be available for two weeks beginning approximately two hours after the call's completion. To access the replay, dial 888-203-1112 (toll-free, domestic only) or 719-457-0820 (domestic and international toll) and enter pass code 2400241.

ASC 606 Implementation

As reported previously, Telenav adopted ASC 606, Revenue from Contracts with Customers, effective July 1, 2018, utilizing the full retrospective transition method. All prior period amounts and disclosures set forth in this earnings release have been adjusted to comply with ASC 606. Under this accounting methodology, certain automotive royalty amounts earned are bifurcated when there exist various underlying obligations. Revenue is recognized upon fulfillment of the underlying obligation. Such various obligations related to earned royalties generally include an onboard navigation component recognized as revenue when each navigation unit is delivered and accepted, a connected services component recognized as revenue over the applicable service period, and a map update component recognized as revenue upon periodic delivery of the applicable map updates.

The adjustments required to transition to ASC 606 on July 1, 2018 resulted in $160.6 million of deferred revenue and $86.9 million of deferred costs originally reported on the company's balance sheet as of June 30, 2018 being recorded instead as revenue and cost of revenue, respectively, in prior periods as adjusted. In addition, the adoption of ASC 606 required the company to capitalize an additional $4.2 million, net, of deferred development costs on its adjusted June 30, 2018 balance sheet, resulting in a net decrease in deferred costs of $82.7 million. The net impact of the Company's adoption of ASC 606 as of June 30, 2018 was an adjustment to decrease its accumulated deficit by $77.8 million. All prior period amounts have been adjusted to comply with ASC 606.

Material Weakness in Internal Control over Financial Reporting

During the three months ended Dec. 31, 2018, Telenav management identified certain errors related to its implementation ASC 606 due to the Company's internal control over financial reporting relating to supervision and review of the financial models supporting Telenav's revenue recognition accounting and disclosures not operating effectively. Telenav management concluded that, because this deficiency created a more than remote likelihood of a material misstatement not being prevented or detected on a timely basis, this deficiency constituted a material weakness in internal control over financial reporting.

A more detailed explanation, together with a description of the remediation plan that we have adopted to address the identified internal control deficiencies, will be included in our Quarterly Report on Form 10-Q for the quarter ended Dec. 31, 2018.

Use of Non-GAAP Financial Measures

Telenav prepares its financial statements in accordance with generally accepted accounting principles for the United States, or GAAP. The non-GAAP financial measures such as billings, direct contribution from billings, direct contribution margin from billings, change in deferred revenue, change in deferred costs, adjusted EBITDA, adjusted cash flow from operations and free cash flow included in this press release are different from those otherwise presented under GAAP. Telenav has provided these measures in addition to GAAP financial results because management believes these non-GAAP measures help provide a consistent basis for comparison between periods that are not influenced by certain items and, therefore, are helpful in understanding Telenav's underlying operating results. These non-GAAP measures are some of the primary measures Telenav's management uses for planning and forecasting. These measures are not in accordance with, or an alternative to, GAAP and these non-GAAP measures may not be comparable to information provided by other companies.

To reconcile the historical GAAP results to non-GAAP financial metrics, please refer to the reconciliations in the financial statements included in this earnings release.

Billings equals revenue recognized plus the change in deferred revenue from the beginning to the end of the applicable period. Direct contribution from billings reflects gross profit plus change in deferred revenue less change in deferred costs from the beginning to the end of the applicable period. Direct contribution margin from billings reflects direct contribution from billings divided by billings. Telenav has also provided a breakdown of the calculation of the change in deferred revenue by segment, which is added to revenue in calculating its non-GAAP metric of billings. In connection with its presentation of the change in deferred revenue, Telenav has provided a similar presentation of the change in the related deferred costs. Such deferred costs primarily include costs associated with third party content and certain development costs associated with its customized software solutions whereby customized engineering fees are earned. As the company enters into more hybrid and brought-in navigation programs, deferred revenue and deferred costs become larger components of its operating results, so Telenav believes these metrics are useful in evaluating cash flows.

Telenav considers billings, direct contribution from billings and direct contribution margin from billings to be useful metrics for management and investors because billings drive revenue and deferred revenue, which is an important indicator of its business. Telenav believes direct contribution from billings and direct contribution margin from billings are useful metrics because they reflect the impact of the contribution over time for such billings, exclusive of the incremental costs incurred to deliver any related service obligations. There are a number of limitations related to the use of billings, direct contribution from billings and direct contribution margin from billings versus revenue, gross profit, and gross margin calculated in accordance with GAAP. First, billings, direct contribution from billings and direct contribution margin from billings include amounts that have not yet been recognized as revenue or cost and may require additional services or costs to be provided over contracted service periods. For example, billings related to certain brought-in solutions cannot be fully recognized as revenue in a given period due to requirements for ongoing map updates and provisioning of services such as hosting, monitoring, customer support, map updates and, for certain customers, additional period content and associated technology costs. Accordingly, direct contribution from billings and direct contribution margin from billings do not include all costs associated with billings. Second, Telenav may calculate billings, direct contribution from billings, and direct contribution margin from billings in a manner that is different from peer companies that report similar financial measures, making comparisons between companies more difficult. When Telenav uses these measures, it attempts to compensate for these limitations by providing specific information regarding billings, direct contribution from billings and direct contribution margin from billings and how they relate to revenue, gross profit and gross margin calculated in accordance with GAAP.

Adjusted EBITDA measures net loss excluding the impact of stock-based compensation expense, depreciation and amortization, other income (expense) net, provision (benefit) for income taxes, and other applicable items such as legal settlements and contingencies, deferred rent reversal and tenant improvement allowance recognition due to sublease termination, net of tax. Stock-based compensation expense relates to equity incentive awards granted to its employees, directors, and consultants. Legal settlements and contingencies represent settlements, offers made to settle, or loss accruals relating to litigation or other disputes in which Telenav is a party or the indemnitor of a party. Deferred rent reversal and tenant improvement allowance recognition represent the reversal of Telenav's deferred rent liability and recognition of Telenav's deferred tenant improvement allowance, as amortization of these amounts is no longer required due to the termination of the company's Santa Clara facility sublease and subsequent entry into a new lease agreement with its landlord for this same facility effective Sept. 2017.

Adjusted EBITDA and adjusted cash flow from operations are key measures used by Telenav's management and board of directors to understand and evaluate Telenav's core operating performance and trends, to prepare and approve its annual budget and to develop short- and long-term operational plans. In particular, Telenav believes that the exclusion of the expenses eliminated in calculating adjusted EBITDA and adjusted cash flow from operations can provide a useful measure for period-to-period comparisons of Telenav's core business.

Adjusted cash flow from operations measures adjusted EBITDA plus the effect of changes in deferred revenue and deferred costs. Telenav believes adjusted cash flow from operations is a useful measure, especially in light of the impact it continues to expect on reported revenue for certain value-added offerings the company provides its customers, including map updates and the impact of future deliverables. Adjusted EBITDA and adjusted cash flow from operations, while generally measures of profitability and the generation of cash, can also represent losses and the use of cash, respectively. In addition, adjusted cash flow from operations is a key financial measure used by the compensation committee of Telenav's board of directors in connection with the development of incentive-based compensation for Telenav's executive officers and employees. Accordingly, Telenav believes that adjusted cash flow from operations generally provides useful information to investors and others in understanding and evaluating Telenav's operating results in the same manner as its management and board of directors.

Free cash flow is a non-GAAP financial measure Telenav defines as net cash provided by (used in) operating activities, less purchases of property and equipment. Telenav considers free cash flow to be a liquidity measure that provides useful information to management and investors about the amount of cash (used in) generated by its business after purchases of property and equipment.

In this press release, Telenav has provided guidance for the third quarter of fiscal 2019 on a non-GAAP basis for billings, direct contribution margin from billings, adjusted EBITDA and adjusted cash flow from operations. Telenav does not provide reconciliations of these forward-looking non-GAAP financial measures to the corresponding GAAP measures due to the high variability and difficulty in making accurate forecasts and projections with respect to deferred revenue, deferred costs, stock-based compensation and tax provision, which are components of these non-GAAP financial measures. In particular, stock-based compensation is impacted by future hiring and retention needs, as well as the future fair market value of Telenav's common stock, all of which is difficult to predict and subject to constant change. The actual amounts of these items will have a significant impact on Telenav's net loss per diluted share and tax provision. Accordingly, reconciliations of Telenav's forward-looking non-GAAP financial measures to the corresponding GAAP measures are not available without unreasonable effort.

Forward Looking Statements

This press release contains forward-looking statements that are based on Telenav management's beliefs and assumptions and on information currently available to its management. Actual events or results may differ materially from those described in this document due to a number of risks and uncertainties. These potential risks and uncertainties include, among others: Telenav's ability to develop and implement products for Ford, GM and Toyota and to support Ford, GM and Toyota and their customers; the impact of Ford's recent announcement regarding the elimination of various sedans in North America and Europe over the near term and GM's recent announcement regarding the elimination of various sedans in North America in the near term; the impact of tariffs on sales of automobiles in the United States and other markets; the impact of the anticipated departure of the United Kingdom from the European Union on sales of automobiles in the United Kingdom and automotive supply chains; Telenav's success in extending its contracts for current and new generation of products with its existing automobile manufacturers and tier ones, particularly Ford; Telenav's ability to achieve additional design wins and the delivery dates of automobiles including Telenav's products; adoption by vehicle purchasers of Scout GPS Link; Telenav's dependence on a limited number of automobile manufacturers and tier ones for a substantial portion of its revenue; reductions in demand for automobiles; potential impacts of automobile manufacturers and tier ones including competitive capabilities in their vehicles such as Apple CarPlay and Android Auto; its advertising business; Telenav's ability to develop new advertising products and technology while also achieving cash flow break even and ultimately profitability in the advertising business; any failure to meet financial performance expectations of securities analysts or investors; failure to reach agreement with customers for awards and contracts on products and services in which Telenav has expended resources developing; competition from other market participants who may provide comparable services to subscribers without charge; the timing of new product releases and vehicle production by Telenav's automotive customers, including inventory procurement and fulfillment; possible warranty claims, and the impact on consumer perception of its brand; Telenav's ability to develop and support products including OpenStreetMap ("OSM"), as well as transition existing navigation products to OSM and any economic benefit anticipated from the use of OSM versus proprietary map products; the potential that Telenav may not be able to realize its deferred tax assets and may have to take a reserve against them; Telenav's reliance on its automobile manufacturers for volume and royalty reporting; the impact on revenue recognition and other financial reporting due to the amendment of contracts or changes in accounting standards; and macroeconomic and political conditions in the U.S. and abroad, in particular China. Telenav discusses these risks in greater detail in "Risk Factors" and elsewhere in its Form 10-Q for the fiscal quarter ended September 30, 2018 and other filings with the U.S. Securities and Exchange Commission ("SEC"), which are available at the SEC's website at www.sec.gov. Given these uncertainties, you should not place undue reliance on these forward-looking statements. Also, forward-looking statements represent management's beliefs and assumptions only as of the date made. You should review the company's SEC filings carefully and with the understanding that actual future results may be materially different from what Telenav expect.

ABOUT TELENAV, INC.Telenav is a leading provider of connected car and location-based services, focused on transforming life on the go for people - before, during, and after every drive. Leveraging our location platform, we enable our customers to deliver custom connected car and mobile experiences. Fortune 500 advertisers and local advertisers can now reach millions of users with Telenav's highly-targeted advertising platform. To learn more about how Telenav's location platform powers personalized navigation, mapping, big data intelligence, social driving, and location-based advertising, visit www.telenav.com.

Copyright 2019 Telenav, Inc. All Rights Reserved.

"Telenav," "Scout," "Thinknear" and the Telenav, Scout and Thinknear logos are registered trademarks of Telenav, Inc. Unless otherwise noted, all other trademarks, service marks, and logos used in this press release are the trademarks, service marks or logos of their respective owners.TNAV-FTNAV-C

Investor Relations:Bishop IRMike Bishop415-894-9633IR@telenav.com

Media:Raphel Finelli408-667-5970raphelf@telenav.com

-- Financial Tables Follow --

Telenav, Inc. Condensed Consolidated Balance Sheets (in thousands, except par value) (unaudited) December 31,2018 June 30,2018As Adjusted [(1)] Assets Current assets: Cash and cash equivalents $ 22,405 $ 17,117 Short-term investments 63,544 67,829 Accounts receivable, net of allowances of $10 and $17 at December 31, 2018 and June 30, 2018, respectively 43,593 46,188 Restricted cash 2,476 2,982 Deferred costs 13,950 11,759 Prepaid expenses and other current assets 3,552 3,867 Total current assets 149,520 149,742 Property and equipment, net 6,396 6,987 Deferred income taxes, non-current 486 867 Goodwill and intangible assets, net 30,479 31,046 Deferred costs, non-current 51,515 46,666 Other assets 3,467 2,372 Total assets $ 241,863 $ 237,680 Liabilities and stockholders' equity Current liabilities: Trade accounts payable $ 22,991 $ 13,008 Accrued expenses 29,367 38,803 Deferred revenue 23,715 20,714 Income taxes payable 258 221 Total current liabilities 76,331 72,746 Deferred rent, non-current 1,051 1,112 Deferred revenue, non-current 64,057 53,824 Other long-term liabilities 993 1,115 Commitments and contingencies Stockholders' equity: Preferred stock, $0.001 par value: 50,000 shares authorized; no shares issued or outstanding -- -- Common stock, $0.001 par value: 600,000 shares authorized; 45,541 and 44,871 shares issued and outstanding at December 31, 2018 and June 30, 2018, respectively 46 45 Additional paid-in capital 170,747 167,895 Accumulated other comprehensive loss (2,010 ) (1,855 ) Accumulated deficit (69,352 ) (57,202 ) Total stockholders' equity 99,431 108,883 Total liabilities and stockholders' equity $ 241,863 $ 237,680 [(1)] Certain amounts have been adjusted to reflect the retrospective adoption of ASC 606. Such amounts were further revised during the three months ended December 31, 2018 to correct certain immaterial errors. Telenav, Inc. Condensed Consolidated Statements of Operations (in thousands, except per share amounts) (unaudited) Three Months Ended Six Months Ended December 31, December 31, 2018 2017As Adjusted [(1)] 2018 2017As Adjusted [(1)] Revenue: Product $ 42,397 $ 45,907 $ 82,327 $ 86,299 Services 14,779 15,492 27,048 29,795 Total revenue 57,176 61,399 109,375 116,094 Cost of revenue: Product 25,015 30,356 48,603 57,679 Services 7,176 7,520 14,350 13,902 Total cost of revenue 32,191 37,876 62,953 71,581 Gross profit 24,985 23,523 46,422 44,513 Operating expenses: Research and development 19,091 21,399 39,193 42,080 Sales and marketing 4,455 5,136 8,870 10,200 General and administrative 5,721 5,514 11,171 10,725 Legal settlements and contingencies 650 60 650 310 Total operating expenses 29,917 32,109 59,884 63,315 Loss from operations (4,932 ) (8,586 ) (13,462 ) (18,802 ) Other income, net 532 218 2,122 171 Loss before provision for income taxes (4,400 ) (8,368 ) (11,340 ) (18,631 ) Provision for income taxes 181 26 811 281 Net loss $ (4,581 ) $ (8,394 ) $ (12,151 ) $ (18,912 ) Net loss per share: Basic and diluted $ (0.10 ) $ (0.19 ) $ (0.27 ) $ (0.43 ) Weighted average shares used in computing net loss per share: Basic and diluted 45,443 44,476 45,230 44,495 [(1)] Certain amounts have been adjusted to reflect the retrospective adoption of ASC 606. Telenav, Inc. Condensed Consolidated Statements of Cash Flows (in thousands) (unaudited) Six Months EndedDecember 31, 2018 2017As Adjusted [(1)] Operating activities Net loss $ (12,151 ) $ (18,912 ) Adjustments to reconcile net loss to net cash used in operating activities: Depreciation and amortization 2,016 1,513 Deferred rent reversal due to lease termination - (538 ) Tenant improvement allowance recognition due to lease termination - (582 ) Accretion of net premium on short-term investments - 113 Stock-based compensation expense 4,384 5,368 Unrealized gain on non-marketable equity investments (1,259 ) - Loss (gain) on disposal of property and equipment (8 ) 6 Bad debt expense 2 37 Changes in operating assets and liabilities: Accounts receivable 2,578 5,545 Deferred income taxes 366 (23 ) Income taxes receivable - 2 Deferred costs (7,040 ) (13,298 ) Prepaid expenses and other current assets 310 (476 ) Other assets 26 (620 ) Trade accounts payable 10,017 (1,563 ) Accrued expenses and other liabilities (9,962 ) (263 ) Income taxes payable 39 (61 ) Deferred rent 89 767 Deferred revenue 13,234 19,840 Net cash provided by (used in) operating activities 2,641 (3,145 ) Investing activities Purchases of property and equipment (446 ) (3,350 ) Purchases of short-term investments (15,862 ) (32,817 ) Proceeds from sales and maturities of short-term investments 20,342 33,322 Net cash provided by (used in) investing activities 4,034 (2,845 ) Financing activities Proceeds from exercise of stock options 26 235 Tax withholdings related to net share settlements of restricted stock units (1,559 ) (1,606 ) Net cash used in financing activities (1,533 ) (1,371 ) Effect of exchange rate changes on cash and cash equivalents (360 ) 563 Net increase (decrease) in cash, cash equivalents and restricted cash 4,782 (6,798 ) Cash, cash equivalents and restricted cash, at beginning of period 20,099 24,158 Cash, cash equivalents and restricted cash, at end of period $ 24,881 $ 17,360 Supplemental disclosure of cash flow information Income taxes paid, net $ 586 $ 640 Reconciliation of cash, cash equivalents and restricted cash to the condensed consolidated balance sheets Cash and cash equivalents $ 22,405 $ 13,956 Restricted cash 2,476 3,404 Total cash, cash equivalents and restricted cash $ 24,881 $ 17,360 [(1)] Certain amounts have been adjusted to reflect the retrospective adoption of ASC 606. Telenav, Inc. Condensed Consolidated Segment Summary (in thousands, except percentages) (unaudited) Three Months Ended Six Months Ended December 31, December 31, 2018 2017As Adjusted [(1)] 2018 2017As Adjusted [(1)] Automotive Revenue $ 47,522 $ 49,157 $ 91,004 $ 92,498 Cost of revenue 28,081 31,981 54,698 60,724 Gross profit $ 19,441 $ 17,176 $ 36,306 $ 31,774 Gross margin 41% 35% 40% 34% Advertising Revenue $ 7,016 $ 8,742 $ 12,963 $ 16,357 Cost of revenue 3,286 4,402 6,506 7,814 Gross profit $ 3,730 $ 4,340 $ 6,457 $ 8,543 Gross margin 53% 50% 50% 52% Mobile Navigation Revenue $ 2,638 $ 3,500 $ 5,408 $ 7,239 Cost of revenue 824 1,493 1,749 3,043 Gross profit $ 1,814 $ 2,007 $ 3,659 $ 4,196 Gross margin 69% 57% 68% 58% Total Revenue $ 57,176 $ 61,399 $ 109,375 $ 116,094 Cost of revenue 32,191 37,876 62,953 71,581 Gross profit $ 24,985 $ 23,523 $ 46,422 $ 44,513 Gross margin 44% 38% 42% 38% [(1)] Certain amounts have been adjusted to reflect the retrospective adoption of ASC 606. Telenav, Inc. Unaudited Reconciliation of Non-GAAP Adjustments (in thousands) Reconciliation of Revenue to Billings Three Months Ended Six Months Ended December 31, December 31, 2018 2017 2018 2017 Automotive Revenue $ 47,522 $ 49,157 $ 91,004 $ 92,498 Adjustments: Change in deferred revenue 6,495 8,940 13,324 20,091 Billings $ 54,017 $ 58,097 $ 104,328 $ 112,589 Advertising Revenue $ 7,016 $ 8,742 $ 12,963 $ 16,357 Adjustments: Change in deferred revenue - - - - Billings $ 7,016 $ 8,742 $ 12,963 $ 16,357 Mobile Navigation Revenue $ 2,638 $ 3,500 $ 5,408 $ 7,239 Adjustments: Change in deferred revenue (103 ) (194 ) (90 ) (251 ) Billings $ 2,535 $ 3,306 $ 5,318 $ 6,988 Total Revenue $ 57,176 $ 61,399 $ 109,375 $ 116,094 Adjustments: Change in deferred revenue 6,392 8,746 13,234 19,840 Billings $ 63,568 $ 70,145 $ 122,609 $ 135,934 Telenav, Inc. Unaudited Reconciliation of Non-GAAP Adjustments (in thousands) Reconciliation of Deferred Revenue to Change in Deferred Revenue Reconciliation of Deferred Costs to Change in Deferred Costs Three Months Ended December 31, 2018 Automotive Advertising Mobile Navigation Total Deferred revenue, December 31 $ 87,325 $ - $ 447 $ 87,772 Deferred revenue, September 30 80,830 - 550 81,380 Change in deferred revenue $ 6,495 $ - $ (103 ) $ 6,392 Deferred costs, December 31 $ 65,465 $ - $ - $ 65,465 Deferred costs, September 30 62,806 - - 62,806 Change in deferred costs $ 2,659 $ - $ - $ 2,659 Three Months Ended December 31, 2017 Automotive Advertising Mobile Navigation Total Deferred revenue, December 31 $ 58,321 $ - $ 633 $ 58,954 Deferred revenue, September 30 49,381 - 827 50,208 Change in deferred revenue $ 8,940 $ - $ (194 ) $ 8,746 Deferred costs, December 31 $ 48,724 $ - $ - $ 48,724 Deferred costs, September 30 43,018 - - 43,018 Change in deferred costs $ 5,706 $ - $ - $ 5,706 Six Months Ended December 31, 2018 Automotive Advertising Mobile Navigation Total Deferred revenue, December 31 $ 87,325 $ - $ 447 $ 87,772 Deferred revenue, June 30 74,001 - 537 74,538 Change in deferred revenue $ 13,324 $ - $ (90 ) $ 13,234 Deferred costs, December 31 $ 65,465 $ - $ - $ 65,465 Deferred costs, June 30 58,425 - - 58,425 Change in deferred costs $ 7,040 $ - $ - $ 7,040 Six Months Ended December 31, 2017 Automotive Advertising Mobile Navigation Total Deferred revenue, December 31 $ 58,321 $ - $ 633 $ 58,954 Deferred revenue, June 30 38,230 - 884 39,114 Change in deferred revenue $ 20,091 $ - $ (251 ) $ 19,840 Deferred costs, December 31 $ 48,724 $ - $ - $ 48,724 Deferred costs, June 30 35,426 - - 35,426 Change in deferred costs $ 13,298 $ - $ - $ 13,298 Telenav, Inc. Unaudited Reconciliation of Non-GAAP Adjustments (in thousands, except percentages) Reconciliation of Gross Profit to Direct Contribution from Billings Three Months Ended Six Months Ended December 31, December 31, 2018 2017 2018 2017 Automotive Gross profit $ 19,441 $ 17,176 $ 36,306 $ 31,774 Gross margin 41% 35% 40% 34% Adjustments to gross profit: Change in deferred revenue 6,495 8,940 13,324 20,091 Change in deferred costs [(1)] (2,659 ) (5,706 ) (7,040 ) (13,298 ) Net change 3,836 3,234 6,284 6,793 Direct contribution from billings [(1)] $ 23,277 $ 20,410 $ 42,590 $ 38,567 Direct contribution margin from billings [ (1)] 43% 35% 41% 34% Advertising Gross profit $ 3,730 $ 4,340 $ 6,457 $ 8,543 Gross margin 53% 50% 50% 52% Adjustments to gross profit: Change in deferred revenue - - - - Change in deferred costs - - - - Net change - - - - Direct contribution from billings $ 3,730 $ 4,340 $ 6,457 $ 8,543 Direct contribution margin from billings 53% 50% 50% 52% Mobile Navigation Gross profit $ 1,814 $ 2,007 $ 3,659 $ 4,196 Gross margin 69% 57% 68% 58% Adjustments to gross profit: Change in deferred revenue (103 ) (194 ) (90 ) (251 ) Change in deferred costs - - - - Net change (103 ) (194 ) (90 ) (251 ) Direct contribution from billings $ 1,711 $ 1,813 $ 3,569 $ 3,945 Direct contribution margin from billings 67% 55% 67% 56% Total Gross profit $ 24,985 $ 23,523 $ 46,422 $ 44,513 Gross margin 44% 38% 42% 38% Adjustments to gross profit: Change in deferred revenue 6,392 8,746 13,234 19,840 Change in deferred costs [(1)] (2,659 ) (5,706 ) (7,040 ) (13,298 ) Net change 3,733 3,040 6,194 6,542 Direct contribution from billings [(1)] $ 28,718 $ 26,563 $ 52,616 $ 51,055 Direct contribution margin from billings [ (1)] 45% 38% 43% 38% [(1)] Deferred costs primarily include costs associated with third party content and in connection with certain customized software solutions, the costs incurred to develop those solutions. We expect to incur additional costs in the future due to requirements to provide ongoing map updates and provisioning of services such as hosting, monitoring, customer support and, for certain customers, additional prepaid content and associated technology costs. Accordingly, direct contribution from billings and direct contribution margin from billings do not reflect all costs associated with billings. Telenav, Inc. Unaudited Reconciliation of Non-GAAP Adjustments (in thousands) Reconciliation of Net Loss to Adjusted EBITDA and Adjusted Cash Flow from Operations Three Months Ended Six Months Ended December 31, December 31, 2018 2017 2018 2017 Net loss $ (4,581 ) $ (8,394 ) $ (12,151 ) $ (18,912 ) Adjustments: Legal settlements and contingencies 650 60 650 310 Deferred rent reversal due to lease termination - - - (538 ) Tenant improvement allowance recognition due to lease termination - - - (582 ) Stock-based compensation expense 2,115 2,888 4,384 5,368 Depreciation and amortization expense 1,006 797 2,016 1,513 Other income, net (532 ) (218 ) (2,122 ) (171 ) Provision for income taxes 181 26 811 281 Adjusted EBITDA (1,161 ) (4,841 ) (6,412 ) (12,731 ) Change in deferred revenue 6,392 8,746 13,234 19,840 Change in deferred costs [(1)] (2,659 ) (5,706 ) (7,040 ) (13,298 ) Adjusted cash flow from operations [(1)] $ 2,572 $ (1,801 ) $ (218 ) $ (6,189 ) [(1)] We expect to incur additional costs in the future due to requirements to provide ongoing map updates and provisioning of services such as hosting, monitoring, customer support and, for certain customers, additional prepaid content and associated technology costs. Accordingly, adjusted cash flow from operations does not reflect all costs associated with billings. Telenav, Inc. Unaudited Reconciliation of Non-GAAP Adjustments (in thousands) Reconciliation of Net Loss to Free Cash Flow Three Months Ended Six Months Ended December 31, December 31, 2018 2017 2018 2017 Net loss $ (4,581 ) $ (8,394 ) $ (12,151 ) $ (18,912 ) Adjustments to reconcile net loss to net cash used in operating activities: Change in deferred revenue [(1)] 6,392 8,746 13,234 19,840 Change in deferred costs [(2)] (2,659 ) (5,706 ) (7,040 ) (13,298 ) Changes in other operating assets and liabilities 2,672 2,260 3,463 3,308 Other adjustments [(3)] 3,110 3,736 5,135 5,917 Net cash provided by (used in) operating activities 4,934 642 2,641 (3,145 ) Less: Purchases of property and equipment (346 ) (1,064 ) (446 ) (3,350 ) Free cash flow $ 4,588 $ (422 ) $ 2,195 $ (6,495 ) [(1)] Consists of product royalties, customized software development fees, service fees and subscription fees. [(2)] Consists primarily of third party content costs and customized software development expenses. [(3)] Consists primarily of depreciation and amortization, stock-based compensation expense and other non-cash items. Telenav, Inc. Summarized Financial Information Depicting the Impact of ASC 606 (in thousands, except per share amounts) (unaudited) As of June 30, 2018 As Reported(ASC 605) Adjustments As Adjusted(ASC 606) Assets Deferred costs $ 31,888 $ (20,129 ) $ 11,759 Deferred costs, noncurrent 109,269 (62,603 ) 46,666 Total assets 320,412 (82,732 ) 237,680 Liabilities and stockholders' equity Deferred revenue 52,871 (32,157 ) 20,714 Deferred revenue, noncurrent 182,236 (128,412 ) 53,824 Accumulated deficit (135,042 ) 77,840 (57,202 ) Total liabilities and stockholders' equity 320,412 (82,732 ) 237,680 Three Months Ended December 31, 2017 Six Months Ended December 31, 2017 As Reported(ASC 605) Adjustments As Adjusted(ASC 606) As Reported(ASC 605) Adjustments As Adjusted(ASC 606) Revenue Product $ 25,307 $ 20,600 $ 45,907 $ 49,271 $ 37,028 $ 86,299 Services 13,773 1,719 15,492 26,467 3,328 29,795 Total revenue 39,080 22,319 61,399 75,738 40,356 116,094 Cost of revenue Product 15,053 15,303 30,356 29,727 27,952 57,679 Services 7,258 262 7,520 13,431 471 13,902 Total cost of revenue 22,311 15,565 37,876 43,158 28,423 71,581 Gross profit 16,769 6,754 23,523 32,580 11,933 44,513 Operating expenses Research and development 21,903 (504 ) 21,399 42,985 (905 ) 42,080 Total operating expenses 32,613 (504 ) 32,109 64,220 (905 ) 63,315 Loss from operations (15,844 ) 7,258 (8,586 ) (31,640 ) 12,838 (18,802 ) Net loss (15,652 ) 7,258 (8,394 ) (31,750 ) 12,838 (18,912 ) Net loss per share, basic and diluted $ (0.35 ) $ 0.16 $ (0.19 ) $ (0.71 ) $ 0.28 $ (0.43 )(C) Copyright 2019 GlobeNewswire, Inc. All rights reserved.

Einstein’s theory of general relativity affords an enormously successful description of gravity. The theory encodes the gravitational interaction in the metric, a tensor field on spacetime that satisfies partial differential equations known as the Einstein equations. This review introduces some of the fundamental concepts of numerical relativity—solving the Einstein equations on the computer—in simple terms. As a primary example, we consider the solution of the general relativistic two-body problem, which features prominently in the new field of gravitational wave astronomy.

The basic equations of general relativity are the Einstein equations, first published in 1915 (1). However, even today there are large gaps in our understanding of the physics implied by the Einstein equations. Stated in general terms, a major goal of research in general relativity is to solve the Einstein equations for the physical situations of interest. Fundamental analytic solutions of the Einstein equations include the flat Minkowski spacetime known from special relativity, the Schwarzschild and Kerr spacetimes describing single black holes, and the simple Big Bang cosmologies. Also predicted by general relativity are gravitational waves, which for weak fields can be obtained as analytic solutions of the linearized Einstein equations. However, the few known analytic solutions describe only very special situations, and approximation methods fail in the regime where the nonlinear, strong-field effects of relativity play a crucial role. If we are interested in the truly relativistic regime, we must turn to computer simulations to obtain numerical solutions to the full Einstein equations.

Solving the full Einstein equations on the computer is the subject of numerical relativity, which could also be called computational general relativity. Computers also play a role in algebraic computations and in approximation schemes, and such calculations are important topics in numerical relativity. But the distinguishing feature of numerical relativity is that, in principle, the Einstein equations in full generality can and must be solved numerically.

Numerical relativity spans a large range of different topics including mathematical general relativity, astrophysics, numerical methods for partial differential equations, computer programming, and simulation science. Current research in numerical relativity is in a transition from a self-contained topic in theoretical physics to a physical theory with numerous connections to observational astronomy (2, 3). Gravitational wave astronomy holds much promise for the future, as recognized by the 2017 Nobel Prize in Physics, and numerical relativity is providing key theoretical predictions and analysis tools for the ongoing gravitational wave observations (4).

The general relativistic two-body problemAs a primary application of numerical relativity, we consider the gravitational two-body problem. The two-body problem in Newtonian gravitational physics can be formulated for two point masses moving in their mutual gravitational field. A particular solution of the Newtonian two-body problem is a Keplerian elliptical orbit. However, in Einsteinian gravity, such orbital motion generates gravitational waves that carry away energy and momentum. Binary orbits therefore decay, and the motion of the two bodies follows an inward spiral that eventually terminates with the collision and merger of the two objects. In most astrophysical situations, the energy loss due to the emission of gravitational waves is so small that a binary orbit decays only on time scales of millions or billions of years. However, for compact objects such as neutron stars or black holes in very tight binaries, general relativistic effects such as gravitational wave emission play a major role (5).

Research in this field seeks to provide a theoretical framework for the physics of binary black holes, neutron stars, and gravitational waves. Such an endeavor must rely on numerical simulations in general relativity and general relativistic hydrodynamics. But a reasonably complete framework still requires substantial progress in numerical relativity and related fields. Currently there are serious limitations in our ability to model the entire range of relevant physics, from the nuclear physics of neutron star matter to the large-scale, strong-gravity effects encountered in binary neutron star mergers (6). The different dynamical phases of the binary evolution—known as the inspiral, the merger, and the evolution of the remnant—are accompanied by characteristic gravitational wave signatures (Fig. 1). For binaries involving at least one neutron star, depending on the specifics of the system, key features include the disruption of the star(s) before merger, the formation of a hypermassive neutron star, the prompt or delayed collapse to a black hole, the dynamics of the accretion torus plus the central merged object, and the creation of unbound material, the ejecta. Before discussing simulations of these systems, we introduce the mathematical foundation of numerical relativity.

Fig. 1 Binary neutron star mergers emit gravitational waves.The waves reveal unique information about extreme gravity and extreme matter—information that can be unraveled with the help of numerical relativity. Shown is a waveform and snapshots of the neutron star matter for the inspiral, merger, and remnant. The amplitude of the gravitational wave is plotted versus time. The merger occurs at t = 0.

IMAGE: COURTESY OF T. DIETRICH, BASED ON (78) Mathematical foundationCombining space and time into spacetime can be considered a triumph of human thought, allowing us to perceive the true nature of relativistic and gravitational physics (7). However, this does not mean that we cannot or need not consider space and time separately. Somewhat ironically, after working hard to unify space and time, the mathematical setup of numerical relativity starts by splitting spacetime again into space and time and by making gauge (coordinate) choices (8) in order to reformulate the Einstein equations as a well-posed mathematical problem.

General relativity is the theory of a metric tensor on a four-dimensional manifold, plus matter described by additional tensor fields. A manifold ℳ endowed with a metric gab is called a spacetime (ℳ, gab). The metric measures lengths, here in four dimensions. The infinitesimal line element

(1)provides a generalization of the Pythagorean theorem. Repeated indices are summed over, following the Einstein summation convention. The metric is symmetric (gab = gba), has a Lorentz signature of – + + +, and there exists an inverse metric gab defined by , where is the identity matrix. A special example is the Minkowski line element ds2 = –c2dt2 + dx2 + dy2 + dz2, where c is the speed of light, t is the time coordinate, and x, y, and z are spatial coordinates. The components of the Minkowski metric are constants, but in general gab is a field with nonconstant components.The field equations of general relativity are the Einstein equations,

(2)where Gab is the Einstein tensor, which depends on the metric and its first and second derivatives, and Tab is the stress-energy tensor constructed from the matter fields Φ and in general also from the metric. For numerical implementations, the first step is to write the Einstein equations as a well-posed system of partial differential equations (PDEs) for the metric. Equation 2 represents 10 coupled, nonlinear PDEs for the 10 independent components of the metric, but without further adjustments these equations are in no known sense hyperbolic (i.e., well-posed as an initial value problem).The differential operator acting on the metric in the Einstein equations is given by the Ricci tensor,

(3)The first term by itself, gcd∂c∂dgab, which is often denoted g□gab (where □ is the d’Alembert operator), has the form of the principal part of a simple hyperbolic wave equation, but note that the metric appears in two places: as the wave field gab and as the inverse metric gcd in the wave operator. The other second-derivative terms are not standard wave operators. The best we can say about the complete principal part in Eq. 3 is that it is quasi-linear in the metric; that is, it is linear in the highest-order derivatives but with coefficients that depend (nonlinearly) on the variable itself. The lower-order terms are quite involved as well, with typical terms of the form g–1g–1∂g∂g. Approaching the problem in this way makes it difficult to recognize that these equations are describing the simple geometric concept of curvature and that there is a time evolution being defined. Further, their well-posedness properties are quite unclear.The so-called 3+1 decomposition—for example, in the form of Arnowitt, Deser, and Misner (ADM) (8)—assumes that the manifold (at least locally) allows a split into time and space, ℳ = R × Σ. Physics is then describable by time-dependent tensors on three-dimensional hypersurfaces Σ, which correspond to t = constant slices of ℳ, resulting in a “foliation” of spacetime in terms of three-dimensional spaces. Geometrically, we obtain a normal vector na to Σ that allows the decomposition of tensors in directions normal and tangential to the hypersurfaces. These are the time-like and space-like directions, respectively. For concreteness, we can assume coordinates xa = (t, xi) = (t, x, y, z) with a time coordinate x0 = t and spatial coordinates xi, where i = 1, 2, 3 and a = 0, 1, 2, 3.

Decomposing the Einstein equations is accomplished by projecting Gab and Rab in the directions parallel and orthogonal to na. We discover that the differential operator Eq. 3 leads to two types of equations: (i) evolution equations containing time derivatives, and (ii) four constraint equations that are essentially elliptic equations, highlighting the indeterminate type of Eq. 3. The constraints are the Hamiltonian constraint and the momentum constraints. The latter are reminiscent of the Gauss law constraint of electrodynamics, where the divergence of the electric field gives the charge density.

Given the evolution and constraint equations as PDEs, we still must choose a spatial and temporal domain with boundary conditions. For problems in astrophysics such as the two-body problem of black holes and neutron stars, we consider isolated systems where at large distances gravitational fields become weak and spacetime becomes asymptotically flat (in contrast to typical cosmological models). Because gravity is universally attractive and long-range, it is not natural to restrict a system to a finite box, especially given that the goal is to compute waves traveling to infinity. Nonetheless, a typical configuration for numerical simulations is a finite-size spatial domain (e.g., a sphere) with boundary conditions at some finite radius that implement the proper fall-off of the fields and an outgoing-wave boundary (9).

Features unique to numerical relativity are various aspects of black hole spacetimes, in particular the causal structure associated with black hole event horizons and the possibility of spacetime singularities. This latter aspect can be viewed as the problem of specifying additional boundaries that represent black holes within the simulation domain.

Building blocks of numerical relativityTo define a particular strategy to solve the Einstein equations, we consider the following building blocks that define the anatomy of a numerical relativity simulation, with a focus on the compact binary problem. The following items are certainly relevant to any evolution problem in computational physics: initial data, evolution, analysis, and numerics. We must specify the initial conditions, integrate the equations of motion to obtain the evolved data, and perform an analysis of the evolved data to extract physical information. The numerical treatment of each item may require the implementation of specific numerical techniques.

EvolutionFormulation: Choose one of many inequivalent formulations (i.e., choose variables and rewrite the Einstein equations to obtain a well-posed initial value problem). Choose the order of time and space derivatives; make structural choices about the gauge and the constraints.

Reformulating the Einstein equations as a well-posed initial value problem has been the subject of much research (10, (11). To give an example, the result of the generalized harmonic gauge (GHG) formulation (12) can be cast in a standard first-order PDE form as

(4)Here, the state vector uμ collects all 10 components gab, the 40 first derivatives ∂cgab, and a few additional fields depending on the formulation. In addition, there may be variables for the matter fields. Equation 4 for GHG is strongly and even symmetric hyperbolic (10, (11). To give an example, the result of the generalized harmonic gauge (GHG) formulation (12) and is well suited for numerical implementation. Another standard way to proceed is the classic ADM formulation that makes the geometry of the 3+1 decomposition in time and space more explicit. Basic variables are the 3-metric gij and the extrinsic curvature Kij, which is essentially the first time derivative of the metric (8). The ADM equations are only weakly hyperbolic and are not suitable for numerics. However, closely related systems, the so-called BSSN and Z4c formulations (10, 11), are strongly hyperbolic. Most current simulations in numerical relativity rely on either the GHG or BSSN/Z4c families of formulations.Constraint propagation: Maintain the constraints during evolution. Perform free evolutions and monitor the convergence of the constraints; use, for example, constraint damping to maintain the constraints explicitly.

Analytically, the constraints propagate; that is, if they are satisfied initially, they remain satisfied during a well-posed evolution. Numerically, even small rounding errors can trigger divergence from the constraint-satisfying solution, which can lead to a catastrophic failure of the simulation. How the constraints are controlled is a distinguishing feature of each formulation. A key ingredient in stable binary black hole evolutions (13) is the constraint-damping scheme (14). The Z4c formulation improves BSSN in the way the Hamiltonian constraint is treated, which leads to improved conservation of mass for neutron star simulations (15). Apart from instabilities, constraint violations in 3+1 relativity signify a problem with four-dimensional covariance. The 3+1 decomposition breaks covariance of the full theory by choosing a foliation, but the constraints ensure that four-dimensional covariance is maintained.

Gauge: Choose a coordinate condition—for example, in terms of lapse and shift or in terms of gauge source functions. Construct coordinates that avoid physical and coordinate singularities and are suitable for the black hole problem.

The main point about the gauge choice is that not only do we have the freedom to choose coordinates, but it is necessary to choose nontrivial coordinates. For example, even for the simplest black hole spacetimes, a foliation can fail by running into the physical singularity, and the hypersurface (or slice) may become badly distorted by slice stretching when points start falling into the black hole. The topic of how to dynamically construct good coordinates that lead to stable evolutions, cover spacetime with a regular foliation, avoid coordinate singularities, and avoid physical singularities inside black holes has become its own area of interest. In that context, the 3+1 decomposition is about “spacetime engineering” because we not only evolve the metric variables, but also build up the spacetime slice by slice in coordinates that are constructed dynamically during the evolution. The GHG formulation relies on the harmonic gauge to obtain hyperbolicity (12). For BSSN, the moving puncture gauge is essential to obtain long-term black hole evolutions, preventing slice stretching (16) and allowing the black hole punctures to move freely (17, 18).

Boundary conditions: Specify outer boundary conditions appropriate for outgoing waves and asymptotic flatness. Specify inner boundaries for black holes; choose between black hole excision and black hole punctures. Handle coordinate patch boundaries.

Although approximate boundary conditions are possible, for a clean treatment, strong or symmetric hyperbolicity is required for well-posedness (11). We can then specify boundary conditions in terms of the ingoing and outgoing characteristic fields. The outer boundary conditions in numerical relativity tend to be substantially more complicated than the Einstein equations themselves, because outgoing-wave boundaries are typically constructed by taking additional derivatives of the right sides of the equations (10). The Einstein equations share with other nonlinear wave equations the feature that there is backscattering by waves off themselves (and, for binary systems, also due to the gradient in the gravitational well). This is a fundamental problem for boundaries at finite radius, because in principle we must account for all future backscattering from outside the domain. Consistent boundaries at finite radius have only been addressed quite recently, considering the long history of the Einstein equations (19, 20).

Initial dataFormulation: Rewrite the constraints as elliptic equations, identifying suitable free and dependent variables.

To give an indication of what the formulation of the constraints entails (8), consider a conformal rescaling of the metric,

, which conveniently transforms the Hamiltonian constraint into a scalar elliptic equation for ψ. We have the freedom to specify a conformal metric, , that is not physical because it does not solve the constraints, but by solving the elliptic equation for ψ we find a physical solution gab that solves the constraints. The conformal transverse-traceless decomposition (8) is widely used for the full set of constraints; for neutron star initial data in particular, the conformal thin-sandwich construction (21, 22) is used, where typically an additional elliptic equation is added to initialize the gauge condition.Physical content: Solve the constraints for data that contain multiple black holes or neutron stars with arbitrary mass, spin, and momentum.

Because the constraints are nonlinear, we cannot simply “add up” the metric tensors of, for example, two Schwarzschild black holes to obtain binary data, although that can be a useful approximate initial guess. As a result, some aspects of the initial data construction are indirect. For example, we can start with two single black hole solutions for particular masses, which will be combined to form a binary. But solving the constraints for the binary leads to a change in the individual masses of the black holes because of the conformal rescaling. In some cases we have to perform evolutions to determine whether the initial data were constructed appropriately for a particular dynamical situation.

There is a growing variety of initial data constructions for binaries that correspond to the variety of physical configurations. For black holes, there are excision-type data, where the interior of black holes is removed (23, 24). Alternatively, black hole puncture data handle the black hole interior with a coordinate singularity at a point (25), which sometimes is called automatic excision. The thin-sandwich formulation is well suited for quasi-equilibrium data of black holes and/or neutron stars, which, for example, can approximate the state of a binary system during a quasi-circular inspiral (26). Only quite recently have methods been developed for neutron stars that generalize the quasi-equilibrium, quasi-circular construction to eccentric orbits (27) and to neutron stars with spin (28) (Fig. 2). Solving the constraints for electromagnetic field configurations is another recent topic of investigation (29).

Fig. 2 Binary neutron star evolution with spin and precession.As a result of the general relativistic frame-dragging effect, a binary of neutron stars with (unaligned) spin will not move within a fixed orbital plane. (A) The orbital motion, indicated by two different colors for the two stars. (B) The angular momentum. Both can show precession and nutation effects, which will also be visible in the gravitational wave signal. The axes indicate spatial coordinates (A) and vector components of the spin (B).

IMAGE: ADAPTED BY C. BICKEL FROM FIGURE 16 OF (79) AnalysisBlack holes and neutron stars: Determine all physical parameters during evolution. Find horizons of black holes. Analyze the rich phenomenology of neutron star mergers with the remnant, torus, jets, and ejecta. Connect to multi-messenger astronomy.

In any binary simulation, a wide range of detailed information is of interest, especially when matter is involved. The “relativity” in general relativity means, however, that many quantities have no direct physical meaning. In general, any tensor component (such as gtt or gxy) is not meaningful by itself; we have to construct proper gauge-invariant quantities. For example, mass and spin must be carefully defined because their local meaning at a point is problematic. For black holes, special methods are required to find the event horizon, which is a global concept in spacetime and therefore expensive to compute. See Fig. 3 for examples. Instead, black hole excision relies on the apparent horizon [e.g., (13)].

Fig. 3 The twisted pair of pants.(A and B) Spacetime plot of the event horizon of two inspiraling black holes that merge and ring down: equal (A) and unequal (B) masses, time t running up, horizontal x-y slices of the event horizon (80). (C) Pair of pants computed numerically in the 1990s for axisymmetric, head-on collisions, time t running up, horizontal slices in z-ρ coordinates (81).

IMAGE: ADAPTED BY C. BICKELGravitational waves: Compute gravitational wave emission; control numerical and systematic errors. Produce gravitational wave templates in a form that is ready to use for gravitational wave detectors. Treat both waveform prediction and waveform analysis.

Gravitational waves are propagating variations in the metric tensor, and the challenge is to separate the physical waves from various coordinate effects. In the weak-field limit, we can define gravitational waves as small perturbations around a background metric, and a first-order gauge-invariant formalism can be used to eliminate leading-order gauge effects (30). Such methods are applicable because we assume that the detectors are located far from the source where an asymptotically flat background is available. In simulations, the numerical grids often include extra patches for the far zone [e.g., (9)], possibly at lower resolution (see below).

A major effort in numerical relativity is directed toward obtaining accurate waveforms with controlled error bars for long time intervals. For the signal-to-noise ratio of current observations, a sufficiently accurate waveform model may begin with a post-Newtonian approximation (assuming nonrelativistic speeds) for the initial inspiral, matched to 10 to 20 orbits up to and including the merger from numerical simulations of the full Einstein equations. Initially, the goal was to filter the signal out of the noise by matching against theoretical waveforms. However, as the quality of the signals is improving, the main goal of gravitational wave astronomy is to estimate unknown source parameters. For example, we need detailed waveform models to distinguish black hole mergers from neutron star mergers, determine masses and spins, etc. The first detection of gravitational waves by Advanced LIGO (2) was accompanied by a theory paper describing how the properties of GW150914 were deduced from the observational data (4). Only by combining data with theory was it possible to arrive at the interpretation of GW150914 as the signature of a binary black hole merger, with specific parameters and credibility intervals. Two families of models were used, the EOBNR and Phenom families of waveforms (2) (Fig. 4). To analyze the data stream from the detectors, various parametrized waveform models are being developed for high-speed template matching (e.g., reduced-order surrogate models) (31).

Fig. 4 Numerical waveform catalogs anticipated the first gravitational wave observations.Shown are examples for template construction for gravitational waves from binary black hole mergers. (A) Various numerical waveforms computed by different research groups forming an international collaboration. (B) Combining post-Newtonian models for the inspiral with numerical relativity. In (A) and (B), the amplitude of the gravitational wave is plotted versus time. The merger occurs at t = 0. In (B), the numerical waveform is preceded by a post-Newtonian waveform to cover more orbits of the inspiral. Such waveforms, which were purely theoretical, became real with the first observation of gravitational waves in 2015 [compare to figure 1 of (2)], making it possible to interpret the first signals as the merger events of two black holes.

IMAGES: (A) ADAPTED BY C. BICKEL FROM FIGURE 1 OF (82); (B) ADAPTED BY C. BICKEL FROM FIGURE 2 OF (83) NumericsDiscretization: Choose a discretization in space and time. Introduce adaptive mesh refinement (AMR) in space and time to efficiently represent different physical length scales. Choose coordinate patches and transformations to adapt coordinates to the underlying physics.

Once a suitably hyperbolic form of the PDEs of general relativity has been derived, we have access to several standard discretizations from applied mathematics. The recent trend has been toward high-order discretizations, with different choices for the geometry and the matter fields. In vacuum or where the matter is smooth, the geometry is smooth as well. For smooth metrics, fourth- to eighth-order finite differencing in space is applied routinely, as well as pseudospectral methods for exponential convergence. Neutron star matter is represented by general relativistic fluids, and handling relativistic shocks becomes important. Several high-resolution shock-capturing (HRSC) fifth-order methods are available (6), as is work on smoothed particle hydrodynamics (32, 33).

The physics of a binary involves several physical scales. The wavelength of gravitational waves near merger is about 100 times the size of the black holes, and the simulation domain is typically chosen to be at least 1000 times the size of the black holes to accommodate several wave cycles. Simulations in three spatial dimensions therefore become several orders of magnitude more efficient with AMR, often of the Berger-Oliger type with refinements not just in space, but also in time. Many codes use several coordinate patches to transition from two (or more) central objects to spherical shells near the outer boundary.

Scientific computing: Implement parallel algorithms for high-performance computing. Invest in professional software engineering for a collaborative computational infrastructure.

Numerical relativity has been very successful with the hybrid MPI (message passing interface) plus OpenMP (open multiprocessing) or a similar parallelization strategy. Still, a typical numerical relativity simulation for a binary coalescence, representing just a single data point in a template catalog, may take roughly 1 month on 1000 to 10,000 cores of a supercomputer. The numerical relativity community is working on improving the efficiency of these methods, including spectral methods and improved AMR schemes, which tend to be a bottleneck for massive parallelism. Most efforts in numerical relativity are group efforts with a long-term investment in an evolving code base. These efforts include SpEC (34), SACRA (35), Whisky/THC (36), Pretorius (37), HAD (38), BAM (39), and the community code Einstein Toolkit (40). Some codes approximate general relativity but provide more advanced neutron star physics (32, 33). Although similar in some regards—after all, the same or similar physics is studied—the different projects vary greatly in the range and the specifics of the physics modules, the flexibility and extensibility of the codes, the level of software optimization, and the collaboration and code-sharing models.

The main challenge common to all these projects is that they are implementing a “moving target,” as formulations and basic equations are still changing and more physics is added to the simulations. Simultaneously, they must handle the trend in technology toward massively parallel computers and heterogeneous hardware, which is challenging given the complex algorithms required for numerical relativity.

Short history of binary simulationsThe first simulations of black holes in vacuum were attempted in 1964 (41). By the 1970s, many concepts of the 3+1 ADM formulation had been brought into numerical relativity (42), which led to the seminal numerical work on head-on (axisymmetric, 2+1-dimensional) black hole collisions and gravitational waves (43, 44). It took until the early 1990s (45, 46) to revisit the head-on collision with improved numerics, which confirmed the early results on gravitational waves (46). Numerical relativity in 3+1 dimensions began in 1995 with the evolution of a Schwarzschild black hole on a Cartesian grid (47) and the evolution of gravitational waves (48), followed by the first fully 3+1-dimensional simulation of a black hole binary (49, 50). All the early black hole simulations mentioned so far were numerically unstable, with barely enough evolution time to start with two separate black holes that promptly merged. The first full orbit was achieved in 2004 (51). In 2005–2006, the last missing ingredients for long-term stable black hole evolutions were found in two different approaches, one based on a harmonic gauge formulation and excision (13) and the other based on the BSSN formulation and black hole punctures (17, 18, 51). By 2010, the robustness and flexibility of these methods had been established. Improvements in the formulations, the boundary conditions, etc., are still ongoing today (11, 12, 51).

Neutron star simulations were pursued in parallel with the black hole simulations. The Valencia formalism of general relativistic hydrodynamics (GRHD), now the primary approach, was developed in the 1990s (52). The first fully general relativistic binary neutron star simulations were published in 2000 (53), with enormous progress in many groups since then. As far as the geometry of general relativity matters in these simulations, it turns out that the methods established for stable black hole simulations carry over to neutron star simulations (gauge, boundaries, initial data formulation, etc.). However, GRHD introduces its own challenge of relativistic shocks, and the range of different physics phenomena makes this a much more complex problem than black holes in vacuum.

OutlookNumerical relativity is developing rapidly in several directions, and we highlight a few representative examples.

High-order methodsHigh-order methods to address the ever-increasing demand for even more accurate and detailed simulations are a major topic of current research. Among the different high-order methods to solve partial differential equations, the discontinuous Galerkin (DG) method has emerged in recent years as a particularly successful general-purpose paradigm (54). It can be argued that the DG spectral-element method subsumes several of the key advantages of traditional finite-element and finite-volume methods. In particular, the DG method works with element-local stencils, which is a great advantage for parallelization and the construction of complicated grids. Furthermore, DG methods offer easy access to hp-adaptivity, where both the size of the computational elements (or cells) and the order of the polynomial approximation within each element can be adapted to the problem.

There are three major efforts to use DG methods for general relativity and/or GRHD (55–57). The first simulations of a single neutron star were achieved recently (55, 58), and simple binaries are a work in progress. With regard to high-order approximations, there is no doubt that if exponentially convergent spectral methods such as DG (or pseudospectral methods) are applicable, they will constitute a big improvement over finite-difference approximations, which give only polynomial convergence. High-order methods can provide breakthroughs by reaching accuracies that make new physics possible (e.g., for magnetic field amplification due to small-scale turbulence) or by reducing numerical errors to make gravitational wave analysis possible. Viewed differently, we can reach a given error criterion with much lower computational resources, making simulations feasible that are otherwise too computationally expensive.

Multi-physicsThe spectacular first observation of both gravitational waves (3) and electromagnetic radiation (59, 60) from a neutron star merger represents the beginning of multi-messenger astronomy including gravitational waves. To model such systems, we need to perform “multi-physics” simulations.

Modeling electromagnetic fields in GRHD can be accomplished by coupling the Maxwell equations to the GRHD equations, for which the prevalent approach has been ideal magnetohydrodynamics (IMHD). The assumption of IMHD is that the fluid has zero resistivity, but for the merger—and in particular for the fields surrounding the remnant with torus and ejecta—the quality of that approximation is unclear. Resistive magnetohydrodynamics (RMHD) is expected to be important for realistic models of plasma instabilities and magnetic reconnection. Apart from unknown physics, the mathematical character of the RMHD equations may be problematic (61, 62). There are only a few general relativistic simulations with RMHD [e.g., (61, 63, 64)]. Developing a proper theory of resistive relativistic plasmas is a large project in itself (65).

The microphysical equation of state of neutron stars remains unknown and is also a target for numerical models and for observations. Investigations may include 20 or more different equations of state in an attempt to cover all sensible proposals. Even determining just one parameter—the existence of neutron stars with 2.0 solar masses (66, 67)—provided a strong constraint. In principle, gravitational wave observations can do much better, gleaning information from the inspiral and the merger. Although inspiral signals will show rather systematic long-time effects (68–70), one of the grand challenges will be to disentangle the much more messy merger signal (71).

Standard merger models predict strong heating of the neutron star matter, which is expected to lead to an enormous amount of neutrino emission with luminosity on the order of 1054 erg s–1. This burst of energy plays a role in models of short gamma ray bursts (72) and also for the ejecta, which in turn affects heavy-element production and macro- or kilonovae (73). However, currently the high dimensionality of such radiative transport problems (3+1 spacetime plus 3 for the radiative transport) is prohibitive, leading to a wide array of approximations with variable applicability (74, 75). A coherent picture for neutrino physics in binary mergers is still lacking but should be a part of multi-messenger astrophysics.

Beyond current astrophysicsNumerical relativity has a large number of applications outside the area of compact binaries and gravitational waves (76, 77). Topics include gravitational collapse with surprising critical phenomena, boson stars and other exotic matter, and cosmological simulations. Going beyond classical general relativity, the field of numerical relativity for alternative gravity theories and gravity in higher dimensions is wide open.

ConclusionThe next decade is sure to see numerical relativity grow in terms of computational power and applicability to different physical scenarios. The detailed theoretical models for black hole and neutron star binaries that are the target of research in numerical relativity are closely linked to the observation of gravitational waves. Numerical relativity, in combination with the highly anticipated future observations of gravitational waves, is expected to provide entirely new insights into extreme gravity and extreme matter.

References and NotesH. Minkowski, in The Principle of Relativity, H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl, Eds. (Dover, 1952), pp. 75–91.

J. W. York Jr., in Sources of Gravitational Radiation, L. Smarr, Ed. (Cambridge Univ. Press, 1979), pp. 83–126.

J. W. York, in Sources of Gravitational Radiation, L. L. Smarr, Ed. (Cambridge Univ. Press, 1979), pp. 83–126.

L. L. Smarr, thesis, University of Texas at Austin (1975).

K. R. Eppley, thesis, Princeton University (1975).

J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods (Springer, 2008).

M. W. Choptuik, L. Lehner, F. Pretorius, in General Relativity and Gravitation: A Centennial Perspective, A. Ashtekar, B. K. Berger, J. Isenberg, M. MacCallum, Eds. (Cambridge Univ. Press, 2015), pp. 361–411.

M. Thierfelder, thesis, University of Jena (2008).

Acknowledgments: I gratefully acknowledge the joint work evident from the list of references. Without my collaborators, this review would not have been possible. Funding: Supported in part by DFG/NSF grant BR 2176/5-1. Author contributions: B.B. is responsible for the entire manuscript. Competing interests: None. Data and materials availability: There are no new data in this review.

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