声波方程隐式高阶有限差分数值模拟

第44卷第2期煤田地质与勘探Vol.44No.22016年4月COALGEOLOGY&EXPLORATIONApr.2016收稿日期:2014-11-06作者简介:(1984—).E-mail:[e-mail]引用格式:.[J].201644(2)106–111.ZHIMin.High-orderimplicitfinitedifferencenumericalsimulationofacousticwaveequation[J].CoalGeology&Exploration201644(2)106–111.文章编号:1001-1986(2016)02-0106-06声波方程隐式高阶有限差分数值模拟智敏(中煤科工集团西安研究院有限公司，陕西西安710077):推导了声波方程空间二阶导数的隐式求解公式及差分系数的求解方法，讨论了该方法的数值频散特征。利用该方法分别对均匀介质及Marmousi模型进行了数值模拟，将其结果与传统的显式差分格式的模拟结果进行了对比分析。结果表明：该方法较传统的显式求解方法具有更低的数值频散、更高的计算精度。：声波方程；数值模拟；隐式差分；数值频散：P631：ADOI:10.3969/j.issn.1001-1986.2016.02.019High-orderimplicitfinitedifferencenumericalsimulationofacousticwaveequationZHIMin(Xi'anResearchInstitute,ChinaCoalTechnologyandEngineeringGroupCorp,Xi'an,710077,China)Abstract:Thepaperderivedimplicitsolutionmethodforsolvingdifferentialequationsandcoefficientofacousticspaceofthesecondderivativeoftheequation,discussednumericalfrequencydispersioncharacteristicsofthismethod.Usingthismethod,Marmousimodelandhomogeneousmediumweresimulatednumerically.Theresultswereanalyzedandcomparedwiththeresultobtainedbytraditionalexplicitdifferencescheme.Theresultshowedthatthismethodwaslowerinnumericaldispersionandmoreaccurateincalculation.Keywords:acousticwaveequation;numericalsimulation;implicitdifference;numericalfrequencydispersion[1-2][3-4][5]BorisBook(FCT)[3-7](2N+2)(4N+2)[8]1方法原理1.1(1)22222221pppxyvt(1)pvxiekxpxxpx2222224sin/2pxpxxpxxpxkxpxxxx(2)22()pxxkx2242224xpppqbxxxx(3)b11,11xxx第2期智敏:声波方程隐式高阶有限差分数值模拟·107·(3)(4)[6]222222222211xpxqbxpxxbxx(4)(4)22222xxqqbxpxx(5)(5)(2)(6)2212xxxbqxxbqxbqxxpx(6)(7)202211Mmmpcpxcpxmxpxmxxx(7)(4)02122cos14sin2Mmmccmkxkxbkx(8)1.2Xkx(8)201122cos2cosMmmXbbXccmX(9)(9)taylor2220101111222!2!nnnMnmnnmXXbccmXnn(10)22011122212111202!21!MMmmmmnnMnnmnmcccmXbcmXnn(11)(11)XXM00120Mmmcc(12)2110Mmmcm(13)211102!21!Mnmmbcmnn(14)(13)—(14)(15)00002222222222221230123612322211232221nnnnMMMMMMMnnMMM212223211200300mcccMcb(15)(15)1,2,,mcmMb(12)0c(6)XZzq(16)21122kkkxztPqqPPv(16)tXkx2M2M+21表1隐式差分格式差分系数表Table1CoefficientsofimplicitdifferenceMbc0c1c2c3c4c5c610.083333–2120.133333–1.70.80.0530.160714–1.490080.6562500.091071–0.00228240.177778–1.345990.5560490.121975–0.0052200.00019050.189394–1.242670.4834810.145503–0.0081540.000526–0.00002160.197802–1.165420.4288380.163854–0.0108660.000946–0.0000680.000003·108·煤田地质与勘探第44卷1.3[9]Clayton-Engquist-majda1.3.1海绵吸收边界条件[9-11]22exp(),0,1,2,,GiaiNbiNb(17)Nb()ia20Nb0.015a1.3.2二阶Clayton-Engquist-majda吸收边界条件Clayton-Engquist-majda222222120PPPvtnvtn(18)n[12]122110,0,1,2,0,1,12211,,1,2,,1,122,0,0,1,(22)21(21)2(22)21(21)2(22)21kkkkkkjxxjxxjxjxjxjkkkkkkNxjxxNxjxxNxjxNxjxNxjxNxjkkkizzizziziPAAPAAPAPAPAPPAAPAAPAPAPAPPAAPAAPAP112,0,112211,,,1,2,,1(21)2(22)21(21)2kkkzizikkkkkkiNzzziNzzzjNzziNzziNzziNzAPAPPAAPAAPAPAPAP(19)/xAvtx/zAvtz2基本流程11Fig.1Flowchartofnumericalsimulationofacousticwaveequation3频散分析(20)fkx101222cos14sin2Mmmccmkxfkxbkxkx(20)22()()4812222Fig.2Dispersioncurvecomparisonofexplicitandimplicitdifferencescheme第2期智敏:声波方程隐式高阶有限差分数值模拟·109·4算例Marmousi4.12200m×2200m5m2000m/s60HzRicker0.5ms(1100m1100m)4833Fig.3Comparisonofwavefieldoftheexplicitandimplicitdifferencescheme44Fig.4Comparisonofwaveformcurveofexplicitandimplicitdifferencescheme5t=700ms65Fig.5Comparisonofwavefieldbeforeandaftersnapshotofboundaryconditions·110·煤田地质与勘探第44卷481222855.462s458.256s22143.376s12124.014s6Fig.6Calculationefficiencyofexplicitandimplicitdifferenceschemes4.2Marmousi7Marmousi7900m3500m5m1500~4800m/s60HzRicker0.5ms3000ms(4250m40m)Z=40mClayton-Engquist-majda20487MarmousiFig.7Marmousimodel8840ms8Fig.8Comparisonofwavefieldofexplicitandimplicitdifferenceschemes9Fig.9Comparisonofsingleshotrecordofexplicitandimplicitdifferencescheme第2期智敏:声波方程隐式高阶有限差分数值模拟·111·9700~850800ms()10Fig.10Comparisonofzerooffsetprofileofexplicitandim-plicitdifferencescheme10()()5结语参考文献[1][J]200921(4)232-236CHENKeyang.High-orderstaggered-gridfinitedifferenceschemeforscalaracousticwaveequation[J].ChinaOffshoreOilAndGas200921(4)232-236.[2][J]201248(36)9-13CAODanping.Multicoreparallelprogrammingoftheacousticequationnumericalmodeling[J].ComputerEngineeringandAp-plications201248(36)9-13.[3][J]200645(2)151-156ZHUShengwangWEIXiucheng.Waveequationmodelingwithimplicitdifferencetechnique[J].GeophysicalProspectingforPe-troleum200645(2)151-156.[4][J]200338(3)247-248CAIQixinHEPeijunQINGuangshengetal.Minimumdis-persionalgorithmbyfinite-differencenumericmodelinganditsapplications[J].OilGeophysicalProspecting200338(3)247-248.[5]FCT[J]201133(12)83-85CHENGJingwangGUHanmingLIULinetal.Fluscorrecti