Приложение.
Определение параметра сходимости.
definition(50,50,50,0.001)
Optimal value of the 'mu' is 1.883000
Решение
основной задачи на сетке
:
У=1 У=0 У=0.1 У=0.7 У=0.6 У=0.5 У=0.4 У=0.3 У=0.2 У=0.9 У=0.8
0 0 0 0 0 0 0 0 0 0 0
0 -0.0013 0.0029 0.0104 0.0207 0.0335 0.0491 0.0680 0.0910 0.1195 0.1564
0 0.0025 0.0125 0.0282 0.0488 0.0743 0.1051 0.1420 0.1866 0.2412 0.3090
0 0.0091 0.0265 0.0511 0.0823 0.1200 0.1650 0.2184 0.2824 0.3598 0.4540
0 0.0173 0.0435 0.0776 0.1192 0.1685 0.2264 0.2945 0.3751 0.4715 0.5878
0 0.0268 0.0626 0.1066 0.1585 0.2186 0.2878 0.3681 0.4621 0.5736 0.7071
0 0.0374 0.0836 0.1379 0.1998 0.2696 0.3484 0.4382 0.5418 0.6637 0.8090
0 0.0492 0.1068 0.1716 0.2432 0.3217 0.4082 0.5045 0.6137 0.7404 0.8910
0 0.0626 0.1327 0.2087 0.2898 0.3761 0.4684 0.5682 0.6783 0.8034 0.9511
0 0.0788 0.1630 0.2507 0.3413 0.4346 0.5312 0.6319 0.7385 0.8544 0.9877
0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000
Модельная задача.
Результат применения функции model(50,50,1.883,10^6):
Maximum error is error = 1.251885121567042e-006
Результат применения функции model(100,100,1.883,10^6):
Maximum error is error = 3.180901020360949e-007
Тексты функций.
function main(M,N,mu,MAX)
% main(M,N,mu,eps,MAX)
% M - the X-size of the grig
% N - the Y-size of the grig
% MAX - the maximum number of the iterations
% mu - the parameter of the convergence (1<mu<2)
warning off
close all
diary d:\main.txt
diary on
format short
a = 1;
b = 1;
h = a/M;
t = b/N;
q = t/h;
eps=((h+t)/2)^4/1.25;
% initialization
u = ones(M+1, N+1);
x = zeros(M+1,1);
y = zeros(N+1,1);
f = -1.*ones(size(u));
% the approximation of the boundary conditions
for i = 1:(M+1),
u(i,1) = 0;
u(i,N+1) = sin(pi*(i-1)*h/2);
x(i) = (i-1)*h;
end
for j = 1:(N+1),
u(1,j) = 0;
u(M+1,j) = (j-1)*t;
y(j) = (j-1)*t;
end
k = 0;
A = 100;
a1 = 1/(2*(1+q^2));
a2 = q^2;
a3 = t^2*a1;
while (A > eps)
k = k + 1;
A = 0;
for i=2:M,
for j=2:N,
X = a1 * (u(i,j-1) + u(i,j+1) + a2*u(i-1,j) + a2*u(i+1,j)) + a3*f(i,j);
X = mu*X + (1-mu)*u(i,j);
R = abs(X - u(i,j));
if (R > A) A = R; end
u(i,j) = X;
end
end
if k == MAX
sprintf('%d steps were made !!!', MAX)
break;
end
end
sprintf('Quality of approximation is %f', A)
if (k ~= MAX)
sprintf('Riched after %d steps', k)
end
m=1;
n=1;
for i=1:M/10:M+1
for j=1:N/10:N+1
result(m,n)=u(i,j);
n=n+1;
end
n=1;
m=m+1;
end
result
U=zeros(M+1,6);
for i=1:M+1,
for j=0:10,
U(i,j+1)=u(i,1+j*N/10);
end
end
i=1:1:M+1;
colormap(pink)
surf(x,y,u);
grid
pause;
colormap('default')
contour(u,50);
pause;
plot(x(i),U)
grid
diary off
function model(M,N,mu,MAX)
% main(M,N,mu,eps,MAX)
% M - the X-size of the grig
% N - the Y-size of the grig
% MAX - the maximum number of the iterations
% mu - the parameter of the convergence (1<mu<2)
warning off
close all
diary d:\model.txt
diary on
format short
a = 1;
b = 1;
h = a/M;
t = b/N;
q = t/h;
eps=((h+t)/2)^4/1.25;
% initialization
u = ones(M+1, N+1);
x = zeros(M+1,1);
y = zeros(N+1,1);
% the approximation of the boundary conditions
for i = 1:(M+1),
u(i,1) = 1;
x(i) = (i-1)*h;
u(i,N+1) = exp(x(i));
end
for j = 1:(N+1),
u(1,j) = 1;
y(j) = (j-1)*t;
u(M+1,j) = exp(y(j));
end
for i=1:(M+1)
for j=1:(N+1)
f(i,j)=-(x(i)^2+y(j)^2)*exp(x(i)*y(j));
U(i,j)=exp(x(i)*y(j));
end
end
k = 0;
A = 100;
a1 = 1/(2*(1+q^2));
a2 = q^2;
a3 = t^2*a1;
while (A > eps)
k = k + 1;
A = 0;
for i=2:M,
for j=2:N,
X = a1 * (u(i,j-1) + u(i,j+1) + a2*u(i-1,j) + a2*u(i+1,j)) + a3*f(i,j);
X = mu*X + (1-mu)*u(i,j);
R = abs(X - u(i,j));
if (R > A) A = R; end
u(i,j) = X;
end
end
if k == MAX
sprintf('%d steps were made !!!', MAX)
break;
end
end
sprintf('Quality of approximation is %f', A)
if (k ~= MAX)
sprintf('Riched after %d steps', k)
end
m=1;
n=1;
for i=1:M/10:M+1
for j=1:N/10:N+1
result(m,n)=u(i,j);
n=n+1;
end
n=1;
m=m+1;
end
result
uu=zeros(M+1,6);
UU=zeros(M+1,6);
for i=1:M+1,
for j=0:10,
uu(i,j+1)=u(i,1+j*N/10);
UU(i,j+1)=U(i,1+j*N/10);
end
end
i=1:1:M+1;
colormap(cool)
surf(x,y,u);
title('obtained')
figure
colormap(pink)
surf(x,y,U);
title('real')
figure
subplot(2,2,1)
plot(x(i),UU,'r')
title('real')
grid
subplot(2,2,2)
plot(x(i),uu,'b')
title('obtained')
grid
subplot(2,2,[3 4])
plot(x(i),UU,'r')
hold
plot(x(i),uu,'b')
grid
title('combined')
R=U-u;
format long
error=max(max(abs(R)));
disp('Maximum error is')
error
diary off
function y = epsilon(M,N,MAX,mu)
% y = epsilon(M,N,MAX,mu)
a = 1;
b = 1;
h = a/M;
t = b/N;
q = t/h;
u = ones(M+1, N+1);
f = -1.*ones(size(u));
x = zeros(M+1,1);
y = zeros(N+1,1);
x = 0:h:1;
y = 0:t:1;
for i = 1:(M+1),
u(i,1) = 0;
u(i,N+1) = sin(pi*(i-1)*h/2);
end
for j = 1:(N+1),
u(1,j) = 0;
u(M+1,j) = (j-1)*t;
end
k = 0;
A = 100;
a1 = 1/(2*(1+q^2));
a2 = q^2;
a3 = t^2*a1;
for k = 1:MAX,
A = 0;
for i=2:M,
for j=2:N,
X = a1*(u(i,j-1)+u(i,j+1)+a2*u(i-1,j)+a2*u(i+1,j))+a3*f(i,j);
X = mu*X + (1-mu)*u(i,j);
R = abs(X - u(i,j));
if (R > A) A = R; end
u(i,j) = X;
end
end
end
y=A;
function definition(M,N,MAX,h)
% y=definition(h,M,N,MAX)
% M - the X-size of the grig
% N - the Y-size of the grig
% MAX - the maximum number of the iterations
% h - the quality of the mu-parameter definition
diary d:\definition.txt
diary on
mu = 1:h:2;
eps = zeros(size(mu));
for i=1:length(mu)
eps(i) = epsilon(M,N,MAX,mu(i));
end
k=0;
EPS=min(eps);
for i=1:length(mu)
if eps(i)==EPS
k=i;
end
end
sprintf('Optimal value of the ''mu'' is %f', mu(k))
diary off