MappedHybridCentral-WENOFiniteDifferenceSchemeforD

摘要：在本项研究中，我们在爆轰波模拟中采用五阶杂交的中心差分和WENO守恒有限差分格式（杂交格式）。杂交格式用来保持解的高梯度和间断的部分一直能被WENO-Z格式用一种本质无震荡方式捕获，而光滑部分将通过有效且精确地中心有限差分格式求解，以加速全部格式的计算。为了探测解的光滑和间断部分，我们使用了Harten提出的一个高阶多分辨率算法。一个正切区域映射被用来将靠近爆轰波前的网格节点聚集起来以增强在半反应区间内的网格分辨率，反应区间将驱动波前后的复杂非线性波动结构的发展。我们对均匀空间网格的WENO-Z格式、WENO-Z格式和区间映射的杂交格式在经典的稳定和不稳定爆轰波模拟方面进行了一些数值比较。一维和二维的数值例子表明Inthisstudy,weemploythefifthorderhybridCentral-WENOconservativefinitedifferencescheme(Hybrid)inthesimulationofdetonationwaves.TheHybridschemeisusedtokeepthesolutionspartsdisplayinghighgradientsanddiscontinuitiesalwayscapturedbytheWENO-Zschemeinanessentiallynon-oscillatorymannerwhilethesmoothpartsarehighlyresolvedbyanefficientandaccuratecentralfinitedifferenceschemeandtospeedupthecomputationoftheoverallscheme.Todetectthesmoothanddiscontinuouspartsofthesolutions,ahighordermulti-resolutionalgorithmbyHartenisused.Atangentdomainmappingisusedtoclustergridpointsnearthedetonationfrontinordertoenhancethegridresolutionwithinhalfreactionzonethatdrivesthedevelopmentofcomplexnonlinearwavestructuresbehindthefront.WeconductseveralnumericalcomparisonsamongtheWENO-Zschemewithauniformlyspacedgrid,theWENO-ZschemeandtheHybridschemewiththedomainmappinginsimulationsofclassicalstableandunstabledetonationwaves.One-andtwo-dimensionalnumericalexamplesshowthattheincreasedgridresolutioninthehalfreactionzonebytheMappedWENO-ZschemeandtheMappedHybridschemeallowsasignificantincreasedefficiencyandaccuracywhencompareswiththesolutionobtainedwithahighlyresolvedonecomputedbytheWENO-Zschemewithauniformlyspacedgrid.Resultsofthree-dimensionalsimulationsofstable,slightlyunstableandhighlyunstabledetonationwavescomputedbytheMappedHybridschemearealsopresented.KeywordsWeightedessentiallynon-oscillatory，Centralfinitedifference，Multi-resolution，Hybrid，Detonationwave在这项研究中，我们采用第五阶混合中心WENO守恒差分格式（混合）在爆轰波模拟。混合方案是用来保持解的部分显示高梯度和不连续性，总是被一个基本无振荡的方式weno-z方案而光滑的部分是高分辨的高效、准确的中心差分格式和加快整体方案的计算。检测解的光滑和连续的部分，是用以Harten高阶多分辨率算法。一个切线域映射的爆前附近的网格点，以提高网格分辨率的半反应区内，驱动开发的复杂的非线性波结构背后的前面。我们进行了一些数值比较均匀间隔的网格之间的weno-zweno-z方案，方案与经典的稳定和不稳定爆轰波模拟域映射的混合方案。一维和二维数值例子表明，增加网格分辨率的半反应区的映射weno-z方案和映射的混合方案可以显著增加时的效率和准确性比一个高度解决一个一个均匀间隔的网格计算解决方案获得weno-z。稳定的，稍微不稳定和高度不稳定的爆轰波计算由映射的混合计划的三维模拟结果。keywordsweighted基本无振荡，中心有限差分，多分辨率，混合，爆轰波1IntroductionDetonationisacomplexphenomenonthatinvolvesashockfrontfollowedbyareactionzone.Itusuallyhappensinsideamineshaftorgastubesinthekitchenandcauseslargepropertydamagesandlossofhumanlives.Thedetailedunderstandingandstudiesonthepropagationofdetonationwaveshaswideapplications.However,directexperimentsofthesephysicalphenomenon,whichexistsinthemostchallengingandhostileenvironment,aredifficultandsafelyconductedifandwhenpossible.Accurateandefficientnumericalsimulationsofamathematicalmodelofdetonationwavesprovideawaytoobtaininsightsinthephysicalproblemsandguideresearcherstohaveadeeperunderstandingofthephysicsandtodesignbetterexperiments.Directnumericalsimulationsoffinescaleanddelicatestructuresofdetonationwavesdemandtheuseofhighlyaccurateandefficientnumericalschemes,whichmustbeabletoresolveaverybroadrangeoflengthandtimescales.Forexample,secondorderGodunovscheme[1,2],extendedspace-timeConservationElement(CE)andSolutionElement(SE)method[3],unsplitscheme[4],classicalWENO-JSscheme[5–7],optimalWENO-Zscheme[8,9]andnon-MUSCL-typeTVDscheme[10]havebeenimplementedtosimulatedetonationwavestoinvestigatethedetonationphenomenonunderdifferentresearchbackgrounds.Inourpreviouswork,thegridconvergencestudyforthecaseofoverdrivefactorf=1.6in[8]showedthattheWENOschemesconvergefasterthanotherexistingnumericalmethodssuchasPPMwithfronttrackingandmeshrefinement[11],unsplitscheme[12],Roe’ssolverwithminmodlimiter[13]andRoe’ssolverwithsuperbeelimiter[13].1引言爆轰是一种复杂的现象，涉及到一个反应区的冲击前。它通常发生在厨房里的一个矿井内或煤气管中，造成巨大的财产损失和生命损失。对爆轰波传播的详细了解和研究有广泛的应用。然而，这些物理现象，在最具挑战性和充满敌意的环境中存在的直接实验，是困难和安全地进行，如果可能的话。准确和有效的数值模拟爆轰波的数学模型提供了一种方法，以获得物理问题的见解，并引导研究人员有一个更深入的了解，并设计更好的实验。直接数值模拟的精细尺度和精细结构的爆轰波需求的高度准确，高效的数值格式，它必须能够解决一个非常广泛的长度和时间尺度的使用。例如，二阶戈杜诺夫格式[1，2]，扩展时空守恒元（CE）和解元（SE）方法[3]，整体方案[4]，经典weno-js方案[5]–7，最佳weno-z方案[8,9]和非MUSCLTVD格式[10]已实施模拟爆轰波研究不同研究背景下的爆震现象。在我们以前的工作中，为加速因子f=1.6in[8]表明，WENO格式收敛比其他现有的数值方法如PPM前跟踪和网格细化[11]的情况下更快的网格收敛性研究，整体方案[12]，Roe求解minmod限[13]和Roe求解器随着SUPERBEE限[13]。Characteristic-basedWeightedEssentiallyNon-Oscillation(WENO)conservativefinitedifferenceschemesasaclassofhighorder/highresolutionmethodforsolutionsofhyperbolicconservationlawsinthepresenceofshocksandsmallscalestructuresinthesolutionwasinitiallydevelopedin[14](fordetailsandhistoryofWENOscheme,see[15]andreferencescontainedtherein).Theuseofadynamicsetofsub-stencilswhereanonlinearconvexcombinationoflowerorderpolynomialsadaptseithertoahigherorderpolynomialapproximationatsmoothpartsofthesolution,ortoalowerorderpolynomialapproximationthatavoidsinterpolationacrossdiscontinuities,yieldsalocalrateofconvergencethatgoesfromorderratthenon-smoothpartsofthesolution,toorder(2r−1)whentheconvexcombinationoflocallowerorderpolynomialsisappliedatsmoothpartsofthesolution.ThenonlinearcoefficientsofWENO’sconvexcombination,referredtoasnonlinearweightsωk,arebasedonlowerorderlo