理論物理中的虛時演化法

1、9 TheequationByemployingimagetimetransformationZ込onecanfindthatitcanbereducedtothefollowingequationdq1dq_z.PWebeginbydiscretizingthedomainXGa、abyplacingagridoverthedomain.Wewillusethegridwithgridspacingh=2a/Ninr.Therefore,ri=a4-(i1)/l(i=12、JV+1).WealsodiscretizingthedomainZG0,6byplacingagridoverthed

2、omainford=b/MinZ.Therefore,Zj=(jI)d(j=1,2,Af4-1)Onecangetthefollowingdifferencescheme9 #9 #namely9 #9 #Bysetting1麗一一=onecanobtain+冋-|a,Qi=甘+%9 #ObviouslyQi=厶君】Qn+i=豈+onecanrewriteitinmetrixformMyL00000QzQiLL00000LX-1L00000LMiL00J+lQd+10.00LM+i00000Lk00000LMv+i丿疔Qn_ikQn/0LUdecompositionWeconsiderLUde

3、compositionasfolowsanauai3a14(000(112131421。22S3a24*21220001U232431。323334*31?3233000134如142QJ3如4丿IU14243144/I0001thenwehave12ai3血4Si22。23。24Q3132。33。34a41a42a43a44丿hl6112hll32122+/21如2bl13+22232114+222432+3112*33+*3113+32“233114+3224+hsU3442+Ml1243+Al13+4223羽+4224+耳3“34丿Thusvveobtain心2=2,34U1J=%=2,

4、3,4*11ii2=心2Al12=2,3,42j=吒f叫j_3,422ii3=心3一Al13一心223=3,4。34一3114一，322434=44=33=a44一4114_42“24_43“34 #21 Ingeneral,wehave #21 # #21 #Wj-i呀-力心m”皿m=li-aij/J,j=2,3,.-,yj=2,3,NJ=久，-VAT-l,j=2+，N #21 # #21 #SolvingasetofequationsSolvingthefollowingequationsAV=DwithA=LU,where000(112u1314212200N=0123243132330

5、00134?414s)k0001丿Forexample,11bl022，324200*33U3000&1000u12100132310(5d】V2必V3dzk如)U14“241 #21 # 21 #Introducingtheintermediatevaraiblesasd1un13W14(vi01“2324V2Z200134V3Z300015)習(xí)in00021/dhi2200Z2d?321330z3血LUdecompositionVeconsiderLUdecompositionasfolows9 #21 #9 #21 #如5213ai4(in000(112131421。2223O24*21

6、330001“34如1如2血3如4丿11414243S/0001thenwehaveana12a13如4S1022。23。24a31。32。33a34如1如2如3如4丿hl6112hll3Gi仏2+，2112：2113+22”232114+22”2432+3112?33+*3113+，3223?3114+，3224+，333442+*411243+4113+4223山4+bl+42“24+耳3“34Thusvveobtain9 #21 #9 #21 #9 #21 #心3厶113心223=3,4。34一/31如4一322433如4一4114Ingeneral,wehav

7、eWj-i呀-力心m”皿m=li-aij/J,j=2,3,.-,yj=2,3,NJ=久，-VAT-l,j=2+，N21 #21 #SolvingasetofequationsSolvingthefollowingequationsAV=DwithA=LU,where000(112u1314212200N=012324313233000134?414s)k0001丿Forexample,11bl022，324200*33U3000&1000u12100132310(5d】V2必V3dzk如)U14“24121 #21 #Introducingtheintermediatevaraiblesasd1un13W14(vi01“2324V2Z200134V3Z300015)習(xí)in00021/dhi2200Z2d?321330z3血3=2,3,Ar-L匕+ij=Lyj=1,2,3,.,A-1,Thesolutionofequationisoftheform習(xí)=#4=J=2,VN=Z2j=Zj-吋J+Wj+I,J=A-1,1summai-ybi00SolvetheequationfvyyC(2=UN丿kd”/5000eg0dma.v/ 21 #wehaveTOC o 1-5 h z5dHl=知12=,21=,hi