一维抛物线型方程数值解法(1)(附图及matlab程序)

一维抛物线偏微分方程数值解法（1）

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, 00)

U(x,0)=e^x, 0<=x<=1,

U(0,t)=e^t,U(1,t)=e^(1+t), 0

精确解为：U(x,t)=e^(x+t);

下面给出两个matlab程序，实质一样（用的是向前欧拉格式）

第二个程序由之前解线性方程组的G-S迭代法得到，迭代次数k=2（固定）

function [p u e x t]=pwxywxq(h1,h2,m,n)

% 解抛物线型一维方程向前欧拉格式（Ut-aUxx=f(x,t),a>0)

%不用解线性方程组，由下一层(时间层）的值就直接得到上一层的值

%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

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));

end

for(i=1:m+1)

u(1,i)=exp(x(i));

end

for(i=1:n+1)

for(j=1:m+1)

f(i,j)=0;

end

end

r=h2/(h1*h1); %此处r=a*h2/(h1*h1）；a=1 要求r<=1/2差分格式才稳定for(i=1:n)

for(j=2:m)

u(i+1,j)=(1-2*r)*u(i,j)+r*(u(i,j-1)+u(i,j+1))+h2*f(i,j);

end

end

for(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));

end

end

或者：

function [u e p x t k]=paowuxianyiweixq(h1,h2,m,n,kmax,ep)

% 解抛物线型一维方程向前欧拉格式（Ut-aUxx=f(x,t),a>0)

%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

syms temp;

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));

end

for(i=1:m+1)

u(1,i)=exp(x(i));

end

for(i=1:n+1)

for(j=1:m+1)

f(i,j)=0;

end

end

a=zeros(n,m-1);

r=h2/(h1*h1);%此处r=a*h2/(h1*h1）；a=1 要求r<=1/2差分格式才稳定

for(k=1:kmax)

for(i=1:n)

for(j=2:m)

temp=(1-2*r)*u(i,j)+r*(u(i,j-1)+u(i,j+1))+h2*f(i,j); a(i+1,j)=(temp-u(i+1,j))*(temp-u(i+1,j));

u(i+1,j)=temp;

end

end

a(i,j)=sqrt(a(i,j));

if(k>kmax)

break;

end

if(max(max(a))

break;

end

end

for(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));

end

end

在命令窗口中输入：

[p u e x t]=pwxywxq(0.1,0.005,10,200);

>> surf(x,t,u)

>> shading interp;

>> xlabel('x');ylabel('t');zlabel('u');

>> title('一维抛物线方程向前欧拉法数值解');

surf(x,t,p)

>> shading interp;

xlabel('x');ylabel('t');zlabel('p');

>> title('一维抛物线方程向前欧拉法精确解')

同理：

plot(x,u)

>> xlabel('x');ylabel('u');

>> title('固定时间改变x u与x 的关系数值解')

[p u e x t]=pwxywxq(0.1,0.01,10,100);

surf(x,t,u)

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

>> surf(x,t,e)

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

Warning: Axis limits outside float precision, use ZBuffer or Painters instead. Not rendering

>>

所以空间步长与时间步长需要满足上面所说的关系