Finite-difference implicit method

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Ewona CZ 2019-12-15

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/496764-finite-difference-implicit-method

I tried to solve with matlab program the differential equation with finite difference IMPLICIT method. The problem: With finite difference implicit method solve heat problem

with initial condition:

and boundary conditions:

. Graphs not look good enough. I believe the problem in method realization(%Implicit Method part).

In the pic above are explicit method two graphs (not this code part here) and below - implicit. I think they shouldn't be like these, they are with hips now, so need help with method realization.

clear;
L = 1.; % Lenth of the wire 0<x<L
T =1; % Number of space steps  0<t<T 
% Parameters needed to solve the equation within the fully implicit methodv
maxk = 1000; % Number of time steps
dt = T/maxk;
n = 10; % Number of space steps
dx = L/n;
a = 1;
b = (a^2)*dt/(dx*dx); % b Parameter of the method
% Initial temperature of the wire:
for i = 1:n+1
    x(i) =(i-1)*dx;
    u(i,1) =sin(x(i));
end
% Temperature at the boundary
for t=1:maxk+1
    time(t) = (t-1)*dt;
    u(1,t) = exp(time(t));
    u(n+1,t) = sin(1)*exp(time(t));
    
end
% Implicit Method 
aa(1:n-1) = -b;
b1=-b;
bb(1:n-1) = 1.+2.*b;
a1=1.+2.*b;
cc(1:n-1) = -b;
c1=-b;
for t = 2:maxk      % Time loop
    uu = u(2:n,t) + dt*(x(2:n)-time(t)).';
    v = zeros(n-1,1); 
    w = a1;
    u(2,t) = uu(1)/w;
    for i=2:(n-1)
        v(i-1) = c1/w;
        w = a1 - b1*v(i-1);
        u(i+1,t) = (uu(i) - b1*u(i,t))/w;
    end
    for j=(n-2):-1:1
        u(j+1,t) = u(j+1,t) - v(j)*u(j+2,t);
    end
    
end
% Graphical representation of the temperature at different selected times
subplot(2,2,3); 
plot(x,u(:,1),'-',x,u(:,10),'-',x,u(:,45),'-',x,u(:,30),'-',x,u(:,60),'-')
title('Temperatura neisreikstiniu metodu')
xlabel('X')
ylabel('T')
subplot(2,2,4); 
mesh(x,time,u')
title('Temperatura neisreikstiniu metodu')
xlabel('X')
ylabel('Temp')