偏微分方程数值解实验报告.doc

.偏微分方程数值解实验报告 一、题目：1、用有限元方法求下列边值问题的数值解：其中其精确解为 ，取h=0.1要求：（1）将精确解与用有限元得到的数值解画在同一图中 （2）、2、 用线性元求解下列问题的数值解：精确到小数点后第六位，并画出解曲面。3、 用Crank-Nicolson差分法求解Burger方程其中取 要求画出解曲面。迭代格式如下：2、 代码：1、%Ritz Galerkin方法求解方程function u1=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for i=1:1:n-1 b(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;for i=1:1:n-1 a(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);hold onplot(x,y1,o)legend(数值解,解析解)function x=yy(a,b,c,d)n=length(b);q=zeros(n,1);p=zeros(n,1);q(1)=b(1);p(1)=d(1);for i=2:1:n q(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);for j=n-1:-1:1 x(j)=(p(j)-a(j)*x(j+1)/q(j);endxx = Columns 1 through 11 0.0157 0.0314 0.0471 0.0628 0.0785 0.0941 0.1097 0.1253 0.1409 0.1564 0.1719 Columns 12 through 22 0.1874 0.2028 0.2181 0.2335 0.2487 0.2639 0.2790 0.2940 0.3090 0.3239 0.3387 Columns 23 through 33 0.3535 0.3681 0.3827 0.3972 0.4115 0.4258 0.4400 0.4540 0.4679 0.4818 0.4955 Columns 34 through 44 0.5091 0.5225 0.5358 0.5490 0.5621 0.5750 0.5878 0.6004 0.6129 0.6253 0.6374 Columns 45 through 55 0.6495 0.6613 0.6730 0.6846 0.6959 0.7071 0.7181 0.7290 0.7397 0.7501 0.7604 Columns 56 through 66 0.7705 0.7805 0.7902 0.7997 0.8090 0.8182 0.8271 0.8358 0.8444 0.8527 0.8608 Columns 67 through 77 0.8687 0.8763 0.8838 0.8910 0.8981 0.9049 0.9114 0.9178 0.9239 0.9298 0.9355 Columns 78 through 88 0.9409 0.9461 0.9511 0.9558 0.9603 0.9646 0.9686 0.9724 0.9759 0.9793 0.9823 Columns 89 through 99 0.9851 0.9877 0.9901 0.9921 0.9940 0.9956 0.9969 0.9981 0.9989 0.9995 0.9999 Column 100 1.0000u = Columns 1 through 11 0.0157 0.0314 0.0471 0.0628 0.0785 0.0941 0.1097 0.1253 0.1409 0.1564 0.1719 Columns 12 through 22 0.1874 0.2028 0.2181 0.2335 0.2487 0.2639 0.2790 0.2940 0.3090 0.3239 0.3387 Columns 23 through 33 0.3535 0.3681 0.3827 0.3972 0.4115 0.4258 0.4400 0.4540 0.4679 0.4818 0.4955 Columns 34 through 44 0.5091 0.5225 0.5358 0.5490 0.5621 0.5750 0.5878 0.6004 0.6129 0.6253 0.6374 Columns 45 through 55 0.6495 0.6613 0.6730 0.6846 0.6959 0.7071 0.7181 0.7290 0.7397 0.7501 0.7604 Columns 56 through 66 0.7705 0.7805 0.7902 0.7997 0.8090 0.8182 0.8271 0.8358 0.8444 0.8527 0.8608 Columns 67 through 77 0.8687 0.8763 0.8838 0.8910 0.8981 0.9049 0.9114 0.9178 0.9239 0.9298 0.9355 Columns 78 through 88 0.9409 0.9461 0.9511 0.9558 0.9603 0.9646 0.9686 0.9724 0.9759 0.9793 0.9823 Columns 89 through 99 0.9851 0.9877 0.9901 0.9921 0.9940 0.9956 0.9969 0.9981 0.9989 0.9995 0.9999 Column 100 1.0000u1 = Columns 1 through 11 0 0.0157 0.0314 0.0471 0.0628 0.0785 0.0941 0.1097 0.1253 0.1409 0.1564 Columns 12 through 22 0.1719 0.1874 0.2028 0.2181 0.2335 0.2487 0.2639 0.2790 0.2940 0.3090 0.3239 Columns 23 through 33 0.3387 0.3535 0.3681 0.3827 0.3972 0.4115 0.4258 0.4400 0.4540 0.4679 0.4818 Columns 34 through 44 0.4955 0.5091 0.5225 0.5358 0.5490 0.5621 0.5750 0.5878 0.6004 0.6129 0.6253 Columns 45 through 55 0.6374 0.6495 0.6613 0.6730 0.6846 0.6959 0.7071 0.7181 0.7290 0.7397 0.7501 Columns 56 through 66 0.7604 0.7705 0.7805 0.7902 0.7997 0.8090 0.8182 0.8271 0.8358 0.8444 0.8527 Columns 67 through 77 0.8608 0.8687 0.8763 0.8838 0.8910 0.8981 0.9049 0.9114 0.9178 0.9239 0.9298 Columns 78 through 88 0.9355 0.9409 0.9461 0.9511 0.9558 0.9603 0.9646 0.9686 0.9724 0.9759 0.9793 Columns 89 through 99 0.9823 0.9851 0.9877 0.9901 0.9921 0.9940 0.9956 0.9969 0.9981 0.9989 0.9995 Columns 100 through 101 0.9999 1.0000ans = Columns 1 through 10 0 0.0157 0.0314 0.0471 0.0628 0.0785 0.0941 0.1097 0.1253 0.1409 Columns 11 through 20 0.1564 0.1719 0.1874 0.2028 0.2181 0.2335 0.2487 0.2639 0.2790 0.2940 Columns 21 through 30 0.3090 0.3239 0.3387 0.3535 0.3681 0.3827 0.3972 0.4115 0.4258 0.4400 Columns 31 through 40 0.4540 0.4679 0.4818 0.4955 0.5091 0.5225 0.5358 0.5490 0.5621 0.5750 Columns 41 through 50 0.5878 0.6004 0.6129 0.6253 0.6374 0.6495 0.6613 0.6730 0.6846 0.6959 Columns 51 through 60 0.7071 0.7181 0.7290 0.7397 0.7501 0.7604 0.7705 0.7805 0.7902 0.7997 Columns 61 through 70 0.8090 0.8182 0.8271 0.8358 0.8444 0.8527 0.8608 0.8687 0.8763 0.8838 Columns 71 through 80 0.8910 0.8981 0.9049 0.9114 0.9178 0.9239 0.9298 0.9355 0.9409 0.9461 Columns 81 through 90 0.9511 0.9558 0.9603 0.9646 0.9686 0.9724 0.9759 0.9793 0.9823 0.9851 Columns 91 through 100 0.9877 0.9901 0.9921 0.9940 0.9956 0.9969 0.9981 0.9989 0.9995 0.9999 Column 101 1.00002、function u = Q_2( P )format longif nargineps %应改为向量2-范数 %构造M for j=1:n2-1 if j=1 M(1,1)=a-y(2);M(1,2)=y(3)/(2*h)-1/(2*h2); end if j=n2-1 M(n2-1,n2-2)=-1/(2*h2);M(n2-1,n2-1)=a-y(n2)/(2*h); end if j=1 & j=n2-1 M(j,j-1)=-1/(2*h2); M(j,j)=a-y(j+1); M(j,j+1)=y(j+2)/(2*h)-1/(2*h2); end end