一维抛物线偏微分方程数值解法(4)(附图及matlab程序)

一维抛物线偏微分方程数值解法（4）上一篇参看一维抛物线偏微分方程数值解法（3）（附图及matlab程序）解一维抛物线型方程(理论书籍可以参看孙志忠：偏微分方程数值解法）Ut-Uxx=0,0x1,0t=1（Ut-aUxx=f(x,t),a0)U(x,0)=e^x,0=x=1,U(0,t)=e^t,U(1,t)=e^(1+t),0t=1精确解为：U(x,t)=e^(x+t);用紧差分格式：此种方法精度为o(h1^2+h2^4),无条件差分稳定；一：用追赶法解线性方程组（还可以用迭代法解）Matlab程序为：function[upext]=JCHGS(h1,h2,m,n)%紧差分格式解一维抛物线型偏微分方程%此程序用的是追赶法解线性方程组%h1为空间步长，h2为时间步长%m,n分别为空间，时间网格数%p为精确解，u为数值解,e为误差x=(0:m)*h1+0;x0=(0:m)*h1;%定义x0,t0是为了f(x,t)~=0的情况%t=(0:n)*h2+0;t0=(0:n)*h2+1/2*h2;symsf;for(i=1:n+1)for(j=1:m+1)f(i,j)=0;%f(i,j)=f(x0(j),t0(i))==0%endendfor(i=1:n+1)u(i,1)=exp(t(i));u(i,m+1)=exp(1+t(i));endfor(i=1:m+1)u(1,i)=exp(x(i));endr=h2/(h1*h1);for(i=1:n)%外循环，先固定每一时间层，每一时间层上解一线性方程组%a(1)=0;b(1)=5/6+r;c(1)=1/12-r/2;d(1)=(r/2-1/12)*u(i+1,1)+...(1/12+r/2)*u(i,1)+(5/6-r)*u(i,2)+(1/12+r/2)*u(i,3)+...h2/12*(f(i,1)+10*f(i,2)+f(i,3));for(k=2:m-2)a(k)=1/12-r/2;b(k)=5/6+r;c(k)=1/12-r/2;d(k)=h2/12*(f(i,k)+...10*f(i,k+1)+f(i,k+2))+(1/12+r/2)*(u(i,k)+u(i,k+2))+(5/6-r)...*u(i,k+1);%输入部分系数矩阵，为0的矩阵元素不输入%一定要注意输入元素的正确性enda(m-1)=1/12-r/2;b(m-1)=5/6+r;d(m-1)=(1/12+r/2)*(u(i,m-1)+u(i,m+1))+...(5/6-r)*u(i,m)+(r/2-1/12)*u(i+1,m+1)+...h2/12*(f(i,m-1)+10*f(i,m)+f(i,m+1));for(k=1:m-2)%开始解线性方程组消元过程a(k+1)=-a(k+1)/b(k);b(k+1)=b(k+1)+a(k+1)*c(k);d(k+1)=d(k+1)+a(k+1)*d(k);endu(i+1,m)=d(m-1)/b(m-1);%回代过程%for(k=m-2:-1:1)u(i+1,k+1)=(d(k)-c(k)*u(i+1,k+2))/b(k);endendfor(i=1:n+1)for(j=1:m+1)p(i,j)=exp(x(j)+t(i));%p为精确解e(i,j)=abs(u(i,j)-p(i,j));%e为误差endend[upext]=JCHGS(0.1,0.005,10,200);surf(x,t,e)title('误差');运行约43秒；[upext]=JCHGS(0.1,0.01,10,100);surf(x,t,e)20多秒；[upext]=JCHGS(0.2,0.04,5,25);surf(x,t,e)3秒；此方法精度很高；二：g-s迭代法求解线性方程组Matlab程序function[uepxtk]=JCFGS1(h1,h2,m,n,kmax,ep)%解抛物线型一维方程格式（Ut-aUxx=f(x,t),a0)%用g-s(高斯-赛德尔)迭代法解%kmax为最大迭代次数%m,n为x,t方向的网格数，例如（2-0）/0.01=200;%e为误差，p为精确解symstemp;u=zeros(n+1,m+1);x=0+(0:m)*h1;t=0+(0:n)*h2;for(i=1:n+1)u(i,1)=exp(t(i));u(i,m+1)=exp(1+t(i));endfor(i=1:m+1)u(1,i)=exp(x(i));endfor(i=1:n+1)for(j=1:m+1)f(i,j)=0;endenda=zeros(n,m-1);r=h2/(h1*h1);%此处r=a*h2/(h1*h1）；a=1for(k=1:kmax)for(i=1:n)for(j=2:m)temp=((1/12+r/2)*(u(i,j-1)+u(i,j+1))+(5/6-r)*u(i,j)+...h2/12*(f(i,j-1)+10*f(i,j)+f(i,j+1))+(r/2-1/12)*(u(i+1,...j-1)+u(i+1,j+1)))/(5/6+r);a(i+1,j)=(temp-u(i+1,j))*(temp-u(i+1,j));u(i+1,j)=temp;%此处注意是u(i+1,j),,而不是u(i+1,j+1)%endenda(i+1,j)=sqrt(a(i+1,j));if(kkmax)break;endif(max(max(a))ep)break;endendfor(i=1:n+1)for(j=1:m+1)p(i,j)=exp(x(j)+t(i));e(i,j)=abs(u(i,j)-p(i,j));endend[uepxtk]=JCFGS1(0.1,0.005,10,200,100000,1e-12);k=67;运行速度1秒左右；surf(x,t,e)[uepxtk]=JCFGS1(0.01,0.001,100,1000,1000000,1e-12);k=5780;surf(x,t,e)