数值分析 课件 chapter8

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1、1lÙ©§ê)©§©n«a.:1)ý.§:X2Poisson§∂2u∂2u2−+=f(x,y),(x,y)∈Ω⊂R∂x2∂y2Ô.§©{V.§©{u(x,y)=ϕ(x,y).ý.§©{∂Ω2)Ô.§:X∂u∂2u¯Ì−=f(x,t),x∈(a,b),t∈(0,T)IK∂t∂x2JJIIu(x,0)=ϕ(x),x∈[a,b],JIu(a,t)=α(t),u(b,t)=β(t),t∈[0,T].11
323)V.§:£22∂u∂u¶w«−=f(x,t),x∈(a,b),t∈(0,T)∂t2∂x2'4u(x,0

2、)=ϕ(x),ut(x,0)=ψ(x),x∈[a,b],òÑu(a,t)=α(t),u(b,t)=β(t),t∈[0,T].1Ô.§©{Äe¡½)¯K:∂u∂2u−a=f(x,t),x∈(0,l),t∈(0,T)(1)∂t∂x2u(x,0)=ϕ(x),x∈[0,l],(2)u(0,t)=α(t),u(l,t)=β(t),t∈[0,T].(3)Ô.§©{V.§©{ý.§©{Ù¥a´~ê§f,ϕ,α,β´®¼ê§÷vϕ(0)=α(0),ϕ(l)=β(0).b½)¯K(1)-(3)3)u(x,t),äk½1

3、w5.¯ÌIK1.1¿©JJIIPD={(x,t)

4、0≤x≤l,0≤t≤T}§h=l/M,τ=T/N,xi=JIih(0≤i≤M),tk=kτ(0≤k≤N),^üq²112
32x=xi,i=0,1,···,M,£t=tk,k=0,1,···,N¶w«'4ò«D©¤Ý/.hÚτ¡mÚÚmÚ.:(x,t)¡!:.PD¯={(x,t)

5、0≤i≤M,0≤k≤N},òÑikhikDh={(xi,tk)

6、1≤i≤M−1,1≤k≤N},Γh=D¯hDh.¡DhSÜ!:§Γh>.!:.3ut=tkþ¤k!:¡1k!:.~

7、^A©úª:1h000g(x0)=[g(x0+h)−g(x0)]−g(ξ1),ξ1∈(x0,x0+h),(4)h21h000g(x)=[g(x)−g(x−h)]+g(ξ),ξ∈(x−h,x),(5)Ô.§©{0002200h2V.§©{ý.§©{1hhh20000g(x0)=gx0+−gx0−−g(ξ3),h2224hh¯Ìξ3∈x0−,x0+,(6)IK22JJII21hJI00(4)g(x0)=h2[g(x0+h)−2g(x0)+g(x0−h)]−12g(ξ4),13
32ξ4∈(x0−h,

8、x0+h),(7)£¶w«1h2g(x)=[g(x−h)+g(x+h)]−g00(ξ),'4000522òÑξ5∈(x0−h,x0+h).(8)1.2;wª(cEulerª)Ä3:(xi,tk)?§∂u∂2u(xi,tk)−a2(xi,tk)=f(xi,tk).(9)∂t∂xd(4)(7)Ô.§©{∂u1τ∂2uV.§©{(x,t)=[u(x,t)−u(x,t)]−(x,η),ý.§©{∂tikτik+1ik2∂t2ikηk∈(tk,tk+1),¯Ì∂2u1(x,t)=[u(x,t)−2u(x,t)+u(

9、x,t)]IK2ik2i+1kiki−1k∂xhJJIIh2∂4uJI−12∂x4(ξi,tk),ξi∈(xi−1,xi+1).14
32£¶w«'4òÑòþ¡üª(9)1[u(xi,tk+1)−u(xi,tk)]τa−2[u(xi+1,tk)−2u(xi,tk)+u(xi−1,tk)]hτ∂2uah2∂4u=f(xi,tk)+(xi,ηk)−(ξi,tk).Ô.§©{2∂t212∂x4V.§©{1≤i≤M−1,0≤k≤N−1.(10)ý.§©{dЩ^(2)Ú>.^(3)u(x,0)=ϕ(x)1≤i≤M

10、−1,¯ÌiiIKu(x0,tk)=α(tk),u(xM,tk)=β(tk),0≤k≤N.JJIIPJIτ∂2uah2∂4u15
32(1)Rik=2∂t2(xi,ηk)−12∂x4(ξi,tk).£¶w«'4òÑ(1)k3(10)¥ÑRik§¿^uiOu(xi,tk)e¡©ªµ1ak+1kkkk(ui−ui)−2(ui+1−2ui+ui−1)=f(xi,tk),τh1≤i≤M−1,0≤k≤N−1,(11)0ui=ϕ(xi),1≤i≤M−1,(12)kku=α(t),u=β(t),0≤k≤N.(13)Ô.§©

11、{0kMkV.§©{ý.§©{(1)©ª(11)*(13)¡;wª(cEulerª)§R¡ikaτ©ª(11)-(13)äØ.Pr=2,¡r´Ú'.(11)h¯ÌIKk+1kkkui=(1−2r)ui+r(ui−1+ui+1)+τf(xi,tk).JJIIJI16
32£¶w«'4òÑOò©ª(11)*