Explicit method for Allen-Cahn equation
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;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; 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; allen cahn equation, nonlinear MATLAB Answers — New Questions
