2、−2222计算(2.2)第一项,有xx11x1⎡⎤⎡⎤duduii++dui+pp−++=∫∫∫22rdxqudx2fdx,(2.3)⎢⎥⎢⎥⎣⎦⎣⎦dx−+11dxxxii−−11dxxi−1ii22222利用Taylor展开式,有11111112311114ux()=−+ux(hhux)=(−+h)hux'(−h)+(hux)''(−+h)(hux)'''(−+hOh)()iiiiii222222!223!2211111112311114u(xhu−=−−)(xhh)=(uxh−+−)(h)'(uxh−)+(−h)u'
3、'(xh−+−)(h)uxhO'''(−+)(h)iiiiii222222!223!22两式做差,整理变形有ux()(ii−−uxh)11231=−+ux'(h)hux'''(−+hOh)()ii2h2223!由此容易验证2323⎡⎤du⎛⎞[][]uuii−−1h⎡⎤du3⎡⎤du⎛⎞[]uuii+1−[]h⎡⎤du3pp=−⎜⎟⎢⎥pO+()h,pp=−⎜⎟⎢⎥pO+()h,⎢⎥13⎢⎥13⎣⎦dxi−1i−2⎝⎠h24⎣⎦dxi⎣⎦dxi+1i+2⎝⎠h24⎣⎦dxi22利用数值积分的中矩形公式,即3bab+−()b
4、a∫f()xdxf=−()(ba)+f′′(),ξξ∈(,)aba224有x1x1i+3i+32qudx=+quhOh[](),2fdx=+fhOh(),∫xii∫xi11i−i−22x1du⎡⎤du[]uu−[]i+33ii+−112rdxr=+hOhr()=+Oh().∫x⎢⎥ii−1dx⎣⎦dxi22将上面的数值积分和数值微分公式代入(2.3)式,得⎛⎞[][]uuii−−−+11⎛⎞[]uuuui[]i[]i+1−[]i−13p⎜⎟−+pr⎜⎟+q[]uh=fh+O()h11iiiiii−+⎝⎠hh⎝⎠222省略掉
5、误差项,用u代替[u],则可定义差分格式为:ii⎡⎤uu−−−uuuuii+−11iiii+1−1−−++⎢⎥pprhqu=hf.11iiiiii+−hh2⎣⎦222对二阶线性椭圆型微分方程⎡⎤∂∂∂∂⎛⎞uu⎛⎞−++⎢⎥⎜⎟pp⎜⎟qu=f(,),xy()x,y∈Ω(2.4)⎣⎦∂∂∂∂xxyy⎝⎠⎝⎠建立差分格式−++++(αuuuuuαααα)=f,(2.5)11ij++,2,1314,1iji−−ij0ijij41111其中ααα01=+=∑kiqpj,221,α2=p1,α31=2p,α4=2p1.试证明差分格
6、式(2.5)的k=1hh12ij++22,,ijhij−,hij,−1222中国地质大学(北京)廉海荣编第二章偏微分方程的有限差分法-2-2局部截断误差的阶为Oh()证明:首先利用二元函数的Taylor展开式,有2⎡⎤⎡⎤∂∂uuh1⎡⎤∂⎡u11∂⎛⎞∂u⎤123⎡⎤∂⎛⎞∂upp=±(,xy)=+ph()±⎢⎥⎜⎟p+()±h⎢⎥⎜⎟p+O(h)⎢⎥⎢⎥ij⎢⎥1121⎣⎦⎣⎦∂∂xx1222⎣⎦∂x⎣⎦∂x⎝⎠∂x!2⎣⎦∂x⎝⎠∂xij±,,ijij,ij,2两式做差,那么有⎡⎤⎡⎤∂∂uupp−⎢⎥⎢⎥⎣⎦⎣⎦∂
7、∂xx11⎡∂⎛⎞∂⎤uij+−22,,ij2⎢⎥⎜⎟pO=+()h1⎣⎦∂∂xx⎝⎠hij,1再由23hh1111⎡⎤∂∂∂uuu1123⎡⎤1⎡⎤4uxhy(,+=+)(ux+=+,y)[u]()h+()h⎢⎥+()h⎢⎥+Oh()iji11j1⎢⎥1231122ij+,2⎣⎦∂x12!2⎣⎦∂∂xx113!2⎣⎦2ij+,ij++,,ij22223hh1111⎡⎤∂∂∂uuu1123⎡⎤1⎡⎤4uxy(,)(ij=+−uxi,)yj=[]uij+1,+(−h11)⎢⎥+(−h)⎢⎥23+−(h1)⎢⎥+Oh()122
8、22⎣⎦∂x12!2⎣⎦∂∂xx113!2⎣⎦ij+,ij++,,ij222可推出[]uu−[]⎡⎤∂∂uuh2⎡⎤3ij+1,i,j13pp=+⎢⎥p+O()h,1⎢⎥231ij+,hx⎣⎦∂123!⎣⎦∂x121ij+,ij+,2233⎡⎤⎡⎤∂∂uu而⎢⎥⎢⎥pp=+O(),h故有331⎣⎦⎣⎦∂
