Matlab:五点差分方法求解椭圆方程非导数边值问题

差分格式脚本文件：

 tic;
 clear
 clc
 M=32;%x的步数
 N=16;%y的步数
 h1=1/M;%x的步长
 h2=1/N;%y的步长
 x=0:h1:1;
 y=0:h2:1;
 u=zeros(M+1,N+1);%给数值解分配内存单元
 U=u;%给精确解分配内存单元
 u(1,:)=y.^3;%y边值
 u(M+1,:)=1+y.^3;%y边值
 u(:,1)=x.^3;%x边值。uo
 u(:,N+1)=1+x.^3;%x边值。un
 for i=1:M+1
     for j=1:N+1
     Accurate(i,j)=x(i)^3+y(j)^3;%精确解
     end
 end
 fun=inline('-6*(x+y)','x','y');
 for i=1:M-1
 for j=1:N-1
         f(i,j)=fun(x(i+1),y(j+1));
 end
 end
 Numerical=u;
 error=eye(M+1,N+1);
  while norm(error,inf) >= 1e-5
 for i=2:M
     for j=2:N
         Numerical(i,j)=(f(i-1,j-1)+N^2*u(i,j-1)+M^2*u(i-1,j)+M^2*u(i+1,j)+N^2*u(i,j+1))/(2*M^2+2*N^2);
     end
 end
  error=Numerical-u;
    u=Numerical;
  end
 [X,Y]=meshgrid(x,y);
 subplot(1,3,1)
 mesh(X,Y,Numerical');
 title('the image of Numerical solution')
 xlabel('x');ylabel('y');zlabel('u');
 subplot(1,3,2)
 mesh(X,Y,Accurate');
 title('the image of Accurate solution')
 xlabel('x');ylabel('y');zlabel('U');
 subplot(1,3,3)
 mesh(X,Y,(Numerical-Accurate)');
 title('the image of error solution')
 xlabel('x');ylabel('y');zlabel('error');
 toc;

效果图：

紧差分格式：

 tic;
 clear
 clc
 M=100;%x的步数
 N=100;%y的步数
 h1=1/M;%x的步长
 h2=1/N;%y的步长
 x=0:h1:1;
 y=0:h2:1;
 u=zeros(M+1,N+1);%给数值解分配内存单元
 U=u;%给精确解分配内存单元
 u(1,:)=y.^3;%y边值
 u(M+1,:)=1+y.^3;%y边值
 u(:,1)=x.^3;%x边值。uo
 u(:,N+1)=1+x.^3;%x边值。un
 for i=1:M+1
     for j=1:N+1
     Accurate(i,j)=x(i)^3+y(j)^3;%精确解
     end
 end
 b1=5/3*(M^2+N^2);
 b2=1/6*(5*M^2-N^2);
 b3=1/6*(5*N^2-M^2);
 f=inline('-6*(x+y)','x','y');
 for i=1:M-1
 for j=1:N-1
         ABf(i,j)=(1/144)*(f(x(i),y(j))+10*f(x(i),y(j+1))+f(x(i),y(j+2))...
             +10*f(x(i+1),y(j))+100*f(x(i+1),y(j+1))+10*f(x(i+1),y(j+2))...
             +f(x(i+2),y(j))+10*f(x(i+2),y(j+1))+f(x(i+2),y(j+2)));
 end
 end
 Numerical=u;
 error=eye(M+1,N+1);
  while norm(error,inf) >= 1e-10
 for i=2:M
     for j=2:N
         Numerical(i,j)=(ABf(i-1,j-1)+1/20*b1*Numerical(i-1,j-1)+b3*Numerical(i,j-1)+1/20*b1*Numerical(i+1,j-1)...
             +b2*Numerical(i-1,j)+b2*u(i+1,j)...
             +1/20*b1*u(i-1,j+1)+b3*u(i,j+1)+1/20*b1*u(i+1,j+1))/b1;
     end
 end
  error=Numerical-u;
    u=Numerical;
  end
 [X,Y]=meshgrid(x,y);
 subplot(1,3,1)
 mesh(X,Y,Numerical');
 title('the image of Numerical solution')
 xlabel('x');ylabel('y');zlabel('u');
 subplot(1,3,2)
 mesh(X,Y,Accurate');
 title('the image of Accurate solution')
 xlabel('x');ylabel('y');zlabel('U');
 subplot(1,3,3)
 mesh(X,Y,(Numerical-Accurate)');
 title('the image of error solution')
 xlabel('x');ylabel('y');zlabel('error');
 toc;

效果图：