偏微分方程数值解抛物型.pdf

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2、First•Prev•Next•Last•GoBack•FullScreen•Close•QuitümÚþh.Kü©ªXXauk+1=buk,β1,β2s+β1,j+β2β1,β2s+β1,j+β2β∈Ω1β∈Ω00us,j=φs,j,s,j=1,2,···,N−1.kkkku0,j=uN,j=us,0=us,N=0,ùpβ=[β1,β2]´þ.Óé½k,g,òÿuk¤½Â3[0,1]×[0,1]s+β1,j+β2þ¼êuk(x,y),d©ªXk+1aβ1,β2u(x+β1h,y+β2h)β∈Ω1Xk=bβ1,β2u(x+β1h,y+β2h)β

3、∈Ω0•First•Prev•Next•Last•GoBack•FullScreen•Close•Quit,òuk(x,y)=vk(m,m)ei(σ1x+σ2y)12þª.ùpm1,m2,σ1,σ2´?¿~ê.-−1XXG(σ,τ)=aeih(β1σ1+β2σ2)·beih(β1σ1+β2σ2)β1,β2β1,β2β∈Ω1β∈Ω0@ok+1kv(m1,m2)=G(σ,τ)v(m1,m2).Ón?Ø,XJDÂÏfk.,@o½5¤á.•First•Prev•Next•Last•GoBack•FullScreen•Close•QuitÞ~:ÄÔ.§Ð>¯K(3.6){w

4、ªuk+1−uks,js,j122
k=2δx+δyus,jτh(3.7)s,j=1,2,···,N−1;k=1,2,···,Nτ−1τùpx,ymÚÑh.-r=2,@oª±=zhk+1kkkkkus,j=(1−4r)us,j+r(us+1,j+us−1,j+us,j+1+us,j−1)òÿ¤ëYª,k•First•Prev•Next•Last•GoBack•FullScreen•Close•Quitk+1ku(x,y)=(1−4r)u(x,y)kk+ru(x+h,y)+u(x−h,y)
kk+u(x,y+h)+u(x,y−h)-uk(x,y)=vk(m,m

5、)eiσ1x+iσ2y,þª,k12k+1kv(m1,m2)=G(σ1,σ2,τ)v(m1,m2),Ù¥,G(σ,σ,τ)=1−4r+r(eiσ1h+e−iσ1h+eiσ2h+e−iσ2h)122σ1h2σ2h=1−4rsin+sin221¦vonNeumann^¤á,r≤.4•First•Prev•Next•Last•GoBack•FullScreen•Close•QuitÄnÔ.§∂u∂2u∂2u∂2u=++···+∂t∂x2∂x2∂x212nÙ{wªuk+1−uk
s1,...,sns,j1222kτ=h2δx1+δx2+···+δxnus1,...,sn

6、s1,...,sn=1,2,···,N−1;k=1,2,···,Nτ−1N´¦ÑÙDÂÏf:G(σ,τ)=1−4r+r(eiσ1h+e−iσ1h+···+eiσnh+e−iσnh)2σ1h2σ2h2σnh=1−4rsin+sin+···+sin222•First•Prev•Next•Last•GoBack•FullScreen•Close•Quitù±,1r≤2n,vonNeumann^¤á,ª½.•First•Prev•Next•Last•GoBack•FullScreen•Close•Quitùp2grN,vonNeumann^3䪽5¥ké/,AO,

7、ª~XêVªÿ,Ù´ª2꽿^.,XJªØ´~Xê,½öØ´ü¹e,vonNeu-mamm^==´7^.=——————————————————————–ª½=⇒vonNeumann^¤á._Ä·K:vonNeumann^ؤá=⇒ªØ½——————————————————————–e¡±_Ä·Kâ?ØRichardsonª(nª)½5.•First•Prev•Next•Last•GoBack•FullScreen•Close•Quit(4)énª