Mtlab:抛物型方程的交替方向隐格式(ADI)

 tic;
 clear
 clc
 M=[,,,,];
 N=M;
 for p=:length(M)
 h=/M(p);% 这里定义空间步长等距
 tau=/N(p); % 时间步长
 x=:h:;
 y=:h:;
 t=:tau:;
 Numerical(M(p)+,M(p)+,N(p)+)=;%u
 numerical(M(p)+,M(p)-)=;%u*
 %-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
 % 求解u*ij和uij过程中构造三对角矩阵
 % a 表示下对角线元素
 % b 表示主对角线元素
 % c 表示上对角线元素
 a=-tau/(*h^)*ones(M(p)-,);
 b=(tau/h^+)*ones(M(p)-,);
 c=-tau/(*h^)*ones(M(p)-,);
 %-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
 for i=:M(p)+
     for j=:M(p)+
         Numerical(i,j,)=exp(/*(x(i)+y(j)));% 初值Numerical（x,y,）=u(i,j,)
     end
 end
 for j=:M(p)+
     for k=:N(p)+
         Numerical(,j,k)=exp(/*y(j)-t(k));%  边值Numerical（,y,t）=u(,j,k)
     end
 end
 for j=:M(p)+
     for k=:N(p)+
         Numerical(M(p)+,j,k)=exp(/*(+y(j))-t(k));%  边值Numerical（,y,t）=u(m,j,k)
     end
 end
 for i=:M(p)+
     for k=:N(p)+
         Numerical(i,,k)=exp(/*x(i)-t(k));%  边值Numerical（x,,t）=u(i,,k)
     end
 end
 for i=:M(p)+
     for k=:N(p)+
         Numerical(i,M(p)+,k)=exp(/*(+x(i))-t(k));%  边值Numerical（x,,t）=u(i,m,k)
     end
 end
 f=inline('-3/2*exp(1/2*(x+y)-t)','x','y','t');
 fun=inline('exp(1/2*(x+y)-t)','x','y','t');
 for i=:M(p)+
     for j=:M(p)+
         for k=:M(p)+
             Accurate(i,j,k)=fun(x(i),y(j),t(k));
         end
     end
 end
 for k=:N(p);
 for j=:M(p)-;% 固定j
 numerical(,j)=-tau/(*h^)*Numerical(,j,k+)+(tau/h^+)*Numerical(,j+,k+)-tau/(*h^)*Numerical(,j+,k+);%  u*0j
 numerical(M(p)+,j)=-tau/(*h^)*Numerical(M(p)+,j,k+)+(tau/h^+)*Numerical(M(p)+,j+,k+)-tau/(*h^)*Numerical(M(p)+,j+,k+);%  u*mj
 for i=:M(p)-
 numerical_right_vector(i,)=tau*f(x(i+),y(j+),t(k)+tau/)+Numerical(i+,j+,k)...
     +tau/(*h^)*(Numerical(i,j+,k)-*Numerical(i+,j+,k)+Numerical(i+,j+,k))...
     +tau/(*h^)*(Numerical(i+,j,k)-*Numerical(i+,j+,k)+Numerical(i+,j+,k))...
     +tau^/(*h^)*(Numerical(i,j,k)-*Numerical(i+,j,k)+Numerical(i+,j,k))...
     +tau^/(*h^)*(-*Numerical(i,j+,k)+*Numerical(i+,j+,k)-*Numerical(i+,j+,k))...
     +tau^/(*h^)*(Numerical(i,j+,k)-*Numerical(i+,j+,k)+Numerical(i+,j+,k));
 end
 numerical_right_vector(,)=numerical_right_vector(,)+tau/(*h^)*numerical(,j);
 numerical_right_vector(M(p)-,)=numerical_right_vector(M(p)-,)+tau/(*h^)*numerical(M(p)+,j);
 numerical(:M(p),j)=chase(a,b,c,numerical_right_vector);
 end
 for i=:M(p)- % 固定i
     for j=:M(p)-
    Numerical_right_vector(j,)=numerical(i+,j);
     end
     Numerical_right_vector(,)=Numerical_right_vector(,)+tau/(*h^)*Numerical(i+,,k+);
     Numerical_right_vector(M(p)-,)=Numerical_right_vector(M(p)-,)+tau/(*h^)*Numerical(i+,M(p)+,k+);
     Numerical(i+,:M(p),k+)=chase(a,b,c,Numerical_right_vector);
 end
 end
 error=abs(Numerical(:,:,M(p)+)-Accurate(:,:,M(p)+));
 error_inf(p)=max(max(error));
 figure(p)
 [X,Y]=meshgrid(y,x);
 subplot(,,)
 surf(X,Y,Accurate(:,:,M(p)));
 xlabel('x');ylabel('y');zlabel('Numerical');
 title('the image of Accurate rusult');
 grid on;
 subplot(,,)
 surf(X,Y,Numerical(:,:,M(p)));
 xlabel('x');ylabel('y');zlabel('Numerical');
 title('the image of Numerical');
 grid on;
 subplot(,,)
 surf(X,Y,error);
 xlabel('x');ylabel('y');zlabel('error');
 title('the image of error Numerical');
 grid on;
 end
 for k=:length(M)
     X=error_inf(k-)/error_inf(k);
 Norm(k-)=log2(X);
 end
 figure(length(N)+)
 plot(:length(N)-,Norm,'-b^');
 xlabel('序号');ylabel('误差阶数');
 title('ADI格式误差阶');
 grid on;
 toc;

取t=1，N=5,10,20,40,80；真解与数值解的结果图：

N=5:

N=10;

N=20;

N=40;

N=80;

误差阶数：