第二章理论基础(5)

§2.4经典差分式介绍。一、双曲型方程经典格式：oxuatu1、迎风格式，(upstream,windward)oxuuaaxuuaatuunjnjnjnjnjnj11122),(..xtoETM.Eq.:33222)1(12612xuccxaxucxaxuatusincos1icccG稳定条件1xtac耗散及频散特性.2,蛙跳格式(LeapFrog)oxuuatuunjnjnjnj221111),(..22xtoET552443322110912016:..xuccxcxucxaxuatuEMsinsin12122iccG1G这是一个无耗散的格式。cccarctgArgGArgGe2sin1sin21223.Lax格式oxuuatuuunjnjnjnjnj21111211txtoET2,.M.Eg.:3322221312xucxaxuccxaxuatusincosicG4.Lax-Wendroff格式njxnjnjnjnjuxtaxuuatuu22211122442333221816:.xuccxaxucxaxuatuEMxkiexkcGGsincos11:25、两步Lax-Wendroff格式step1:oxuuauuunjnjtnjnjnj12122121step2:oxuuatuunjnjnjnj21212121116,MacCormack显式格式step1:(Predictstep)njnjnjjuuxtauu1step2:(Correctstep)*1****jjjjuuxtauu**211jnjnjuuu作业*分别针对标准的模型方程与非线性方程（例如Eueler方程）讨论Lax-Wendroff,两步L-W格式和MacCormack格式的异同,7,Beam-Warming两步迎风格式(a0)P:njnjnjjuuxtauu1*C:njnjnjjjjnjnjuuuxtauuxtauuu21*1**2112该格式合并一步的形式是：;njnjnjnjnjnjnjuuuccuucuu21211121或njnjnjnjnjnjnjuuuxcaxuuatuu2111221M,E：4424332218216xuccctxxuccxaxuaxu稳定条件oc22c2sin121sin2sin2sin1221222xkcxkicxkxkcccG8,隐式格式;0211111xuuatuunjnjnjnj332322223621:,xutaxaxutaxuatuEgM22sin1sin1cicG分析与思考：隐式格式的稳定性如何。9,时间中心隐式格式(梯形差分格式)Time-CenteredimplicitMethod(TrapezoidalDifferencingMethod)由Taylor’展开;1623332221njnjnjnjnjtuttuttutuu2621333122211njnjnjnjnjtuttuttutuu同时考虑123322122ttuttutunjnjnj(1)-(2)：)(2311totututuunjnjnjnj对于xuatuoxuatu3112toxuaxuatuunjnjnjnj若xu采用中心差njnjnjnjnjnjuuuucuu1111111455442234332238024120612:,xutaxtataxuxataxuaxuEgMsin21sin21icicG无条件稳定；但当C很大时,1G解是振荡的10，介绍一种高阶格式，Rusanov(Burstein-Mirin)Mefhgd这是一种三步格式njnjnjnjjuucuuu11132121121121232jjnjjuucuu21212112183277224jjnjnjnjnjnjnjuucuuuucuunjnjnjnjnjuuuuu211246424稳定条件：c3442cc55424443341545120424:.xuccxaxucccxaxuatuEgM当424cc时无4阶粘性项当541422cc时，无5阶频散项。2sin1321sin2sin32sin2122422ciccG二，抛物型方程经典格式（通过模型方程22xuatu来描述）1、显式FTCS格式方程：22xuaunjxnjnjuxatuu2216642212123314422212236012:..xxaxtataxuxataxuatuEqMu（注意由上述M。Eq。不能直接看出稳定条件,）1cos21xksG2．混合(显－隐)格式.njxnjxnjnjuuxatuu212211oa当时为FTCS1当b为隐式格式时格式为，当NicolsonCrankc216642223244222236012161311221:..xuxaxtataxuxataxuatuEqM21i22,xtoETstaxii121211221242,,xtoET201,12121ssiii63,,xtoET稳定性分析;按Von-Neumam稳定性分析方法,可得2sin412sin14122xksxksGoos上式中分母恒大于零G12sin412sin14122xksxksG解上述不等式；2sin412sin14122xksxks2sin412sin1412sin41222xksxksxks右边；22sin142sin422xksxks12sin2)21(2xks讨论；21若i上式无条件满足无条件稳定21iissxks412121212sin21212或或则稳定条件综合格式之图示如下：s121211/21/61/2s4121s203,DuFort-Frankel格式、sRichardson'*格式；2111122xuuutuunjnjnjnjnj这是绝对不稳定的格式！修正为；21111112xuuuutuunjnjnjnjnjnjD-F格式njnjnjnjnjuuusuus11111221仍为显式格式讨论稳定性；xksssG21sin41cos222若os22sin41则：若os22sin41则ssisG211sin4cos22212114211sin4cos421)1sin4(cos22222222222222ssssssssG稳定性分析结论是――无条件稳定的。但是对DuFot-Frankel格式的相容性分析,(或推导M.Eg.)有:)(,)(,..2322xtxttoET或66442344422322223360112.;.xuxtaxaxxuxtaxaxuatuEgMs由此可见DnFort-Franew格式的相容性是有条件的，1441sin41cos42cos4121sin41cos222222222ssssssssG即必须有0lim00xtxt才是相容的，即t应是x的高阶小量，这对计算是附加了一个严格的要求！为什么？固而对于使用该格式而言，对时间步长仍有限制，虽然不是从稳定性的要求提出的！4,Box(盒式)格式不等距网格划分的Box格式：方程：22xuxu令xuvvxuxvtu)1(2111111121njnjnjjnjnjvvvxuujnjnjnjnjjnjnjnnjnjnnjnjnjnjxvvvvxvvtuutuuuu2211111111111121212121)2(1111111111jnjnjnnjnjjnjnjnnjnjxvvtuuxvvtuu由（1）式得；njjnjnjnjnjjnjnjnjnjjnjnjnjnjjnjnjnjvxuuvvxuuvvxuuvvxuuv1111111111111111)(2)(2)(2)(2上式两组关系分别代入（2）式，分别得到（3）和（3）’式：)3(22222111211111111jnjnjjnjnnjnjjnjnjjnjnnjnjxuuxvtuuxuuxvtuu)'3(2222211112111111111jnjnjjnjnnjnjjnjnjjnjnnjnjxuuxvtuuxuuxvtuu将（3）’式之下标增加1、即1j改为j，余类推，得：)4(222221111121111111111jnjnjjnjnnjnjjnjnjjnjnnjnjxuuxvtuuxuuxvxuu:;1)4(1)3(11得和消去njnjjjvvxxnjnjjnjjnjjCuAuDuB11111其中111122jnjjjnjjxtxAxtxB111122jjnjnjjxxtxtxD1111111122