Explicit method for Allen-Cahn equation

12 次查看（过去 30 天）
Erm
Erm 2024-10-1
评论： Erm 2024-10-1
Ran in:
The plot of the equation must start at x=-1 and end at x=1. but mu result did not show that?
clear all;
clc;
maxk = 1000;
T = 0.10;
n = 50;
L = 2; % Length of the spatial domain [−1, 1]
Nx = 400; % Number of spatial grid points
dx = L / (Nx - 1); % Spatial step size
dt = T/maxk;
T = 1; % Final time
Nt = round(T / dt); % Number of time steps
a = 0.0001;
r = a * dt / (dx * dx); % Diffusion factor for explicit scheme
% Initial condition
x = linspace(-1, 1, n+1);
u = zeros(n+1, maxk+1);
u(:,1) = x.^2 .* cos(pi * x);
% Implementation of the explicit method for Allen-Cahn equation
for t = 1:maxk
% Internal points
for i = 2:n
u(i, t+1) = u(i, t) + r * (u(i-1, t) - 2 * u(i, t) + u(i+1, t)) ...
+ dt * (5 * u(i, t)^3 - 5 * u(i, t));
end
% Periodic boundary conditions
u(1, t+1) = u(end-1, t+1); % Periodic condition for first point
u(end, t+1) = u(2, t+1); % Periodic condition for last point
end
% Plot results
figure; % Create a new figure
xx = linspace(-1, 1, 100);
t_values = [0, 0.2, 0.4, 0.6, 0.8]; % Time values to plot
plot(x, u(:,1), '-', x, u(:,round(maxk*0.2)), '-', x, u(:,round(maxk*0.4)), '-', x, u(:,round(maxk*0.6)), '-', x, u(:,end), '-');
xlabel('x');
ylabel('u(x,t)');
grid on;
legend('t = 0', 't = 0.2', 't = 0.4', 't = 0.6', 't = 0.8');
hold off;
  1 个评论
Torsten
Torsten 2024-10-1
Did you read somewhere that you need a boundary condition on the gradients of u ? In my opinion, u(-1,t) = u(1,t) suffices to fix a solution.

请先登录，再进行评论。

请先登录，再回答此问题。

采纳的回答

Torsten
Torsten 2024-10-1
Ran in:
xstart = -1.0;
xend = 1.0;
nx = 401;
x = linspace(xstart,xend,nx).';
dx = x(2)-x(1);
tstart = 0.0;
tend = 1.0;
nt = 10;
tspan = linspace(tstart,tend,nt);
u0 = x.^2.*cos(pi*x);
D = 1e-4;
[T,U] = ode15s(@(t,u)fun(t,u,D,nx,dx),tspan,u0);
plot(x,[U(1,:);U(end,:)])
function dudt = fun(t,u,D,nx,dx)
ufull = [u(end-1);u;u(2)];
dudt = D*(ufull(3:end)-2*ufull(2:end-1)+ufull(1:end-2))/dx^2-5*ufull(2:end-1).^3+5*ufull(2:end-1);
end
  4 个评论
显示 2更早的评论
Torsten
Torsten 2024-10-1
编辑：Torsten 2024-10-1
On your code why you wrote D = 1e-4
I prefer the scientific notation for small numbers. Think about how many 0's you had to write if D was 1e-10 instead of 1e-4 :-)

请先登录，再进行评论。

更多回答（0 个）

类别

在 Help Center 和 File Exchange 中查找有关 Get Started with Curve Fitting Toolbox 的更多信息

标签

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by