helmholtz.m

这是一类特殊的椭圆偏微分方程Helmhotz方程。该程序是用迭代法求解Helmhotz方程的。

function [u,x,y] = Helmholtz(f,g,bx0,bxf,by0,byf,D,Mx,My,MinErr,MaxIter)
%解方程： u_xx + u_yy + g(x,y)u = f(x,y)
% 自变量取值区域 D = [x0,xf,y0,yf] = {(x,y) |x0 <= x <= xf, y0 <= y <= yf}
% 边界条件
% u(x0,y) = bx0(y), u(xf,y) = bxf(y)
% u(x,y0) = by0(x), u(x,yf) = byf(x)
% x轴均分为Mx段
% y轴均分为My段 
% tol 误差因子
% MaxIter: 最大迭代次数
x0 = D(1); xf = D(2); y0 = D(3); yf = D(4);
dx = (xf - x0)/Mx; x = x0 + [0:Mx]*dx;%构造内点数组
dy = (yf - y0)/My; y = y0 + [0:My]'*dy;
Mx1 = Mx + 1; My1 = My + 1;
%边界条件
for m = 1:My1
    u(m,[1 Mx1])=[bx0(y(m)) bxf(y(m))]; %左右边界
end
for n = 1:Mx1
    u([1 My1],n) = [by0(x(n)); byf(x(n))];%上下边界
end 
%边界平均值作迭代初值
sum_of_bv = sum(sum([u(2:My,[1 Mx1]) u([1 My1],2:Mx)']));
u(2:My,2:Mx) = sum_of_bv/(2*(Mx + My - 2));
for i = 1:My
    for j = 1:Mx
         F(i,j) = f(x(j),y(i)); G(i,j) = g(x(j),y(i));
    end
end
dx2 = dx*dx; dy2 = dy*dy; dxy2 = 2*(dx2 + dy2);
rx = dx2/dxy2; ry = dy2/dxy2; rxy = rx*dy2;
for itr = 1:MaxIter
    for j = 2:Mx
        for i = 2:My
            u(i,j) = ry*(u(i,j + 1)+u(i,j - 1)) + rx*(u(i + 1,j)+u(i - 1,j))+ rxy*(G(i,j)*u(i,j)- F(i,j)); %迭代公式
        end
    end
    if itr > 1 & max(max(abs(u - u0))) < MinErr%循环结束条件
        break; 
    end
    u0 = u;
end