偏微分方程数值解实验报告2)u-uh'Himax1 1Vn2-2Vn+1+Vnj1+h2Vn+1—2Vn+1+Vn+1Ji1jJ—L>h2一、题目：1、用有限元方法求下列边值问题的数值解兀 2兀 x-y'+y=2sin(),00,xe(0,1),te(0,5),txxxv(x,0)=sin2 兀 x,xe[0,l];v(0,t)=v(1,t)=0.其中取 v=1要求画出解曲面。迭代格式如下：Vn+1—Vn1[(Vn)2—(Vn)2(Vn+1)2-(Vn+1)21-4j-+—<—j—+—j>T4hh二、代码：1、%RitzGalerkin 方法求解方程functionu1=Ritz(x)%定义步长h=1/100;x=0:h:1;n=1/h;a=zeros(n-1,1);b=zeros(n,1);c=zeros(n-1,1);d=zeros(n,1);%求解 Ritz 方法中内点系数矩阵fori=1:1:n-1b(i)=(1/h+h*pi*pi/12)*2;d(i)=h*pi*pi/2*sin(pi/2*(x(i)+h))/2+h*pi*pi/2*sin(pi/2*x(i+1))/2;end%右侧导数条件边界点的计算b(n)=(1/h+h*pi*pi/12);d(n)=h*pi*pi/2*sin(pi/2*(x(i)+h))/2;fori=1:1:n-1a(i)=-1/h+h*pi*pi/24;c(i)=-1/h+h*pi*pi/24;end%调用追赶法u=yy(a,b,c,d)%得到数值解向量u1=[0,u]%对分段区间做图plot(x,u1)%得到解析解y1=sin(pi/2*x);holdonplot(x,y1,'o')legend('数值解','解析解')functionx=yy(a,b,c,d)n=length(b);q=zeros(n,1);p=zeros(n,1);q(1)=b(1);p(1)=d(1);fori=2:1:nq(i)=b(i)-a(i-1)*c(i-1)/q(i-1);p(i)=d(i)-p(i-1)*c(i-1)/q(i-1);endx(n)=p(n)/q(n);forj=n-1:-1:1x(j)=(p(j)-a(j)*x(j+1))/q(j);endxx=Columns1through110.01570.03140.04710.06280.12530.14090.15640.17190.07850.09410.1097Columns12through220.18740.20280.21810.23350.29400.30900.32390.33870.24870.26390.2790Columns23through330.35350.36810.38270.39720.45400.46790.48180.49550.41150.42580.44000.01570.03140.04710.06280.07850.09410.1097Columns34through440.50910.52250.53580.54900.56210.57500.58780.60040.61290.62530.6374Columns45through550.64950.66130.67300.68460.72900.73970.75010.76040.69590.70710.7181Columns56through660.77050.78050.79020.79970.83580.84440.85270.86080.80900.81820.8271Columns67through770.86870.87630.88380.89100.91...
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