Hello everyone! I am new to FreeFEM++ and I am working on solving the following partial differential equation (PDE):
x^2 (1 - x - y)^2 \frac{\partial^2 f}{\partial x^2} + (\beta (1 - x - y)-1) x \frac{\partial f}{\partial x} + \gamma x \frac{\partial f}{\partial y} - \epsilon f - \frac{\partial f}{\partial \tau} = 0
Initial Condition :
f(x, y, 0) = \max(x - k, 0)
Boundary Condition :
Neumann-Dirichlet boundary condition: f(0, y, \tau) = 0
Where 0 \leq x + y \leq 1 , and x, y \in [0, 1] .
When I run the following code on a Mac, I encounter an error:
before token ) in the parallel computation part of the weak form.
I am not sure what is causing the issue, especially in the parallel computation section. Could someone assist me in identifying the problem and suggest how to fix it?
// Load necessary libraries
load "msh3"
load "isoline"
// Parameter settings
real beta = 0.5;
real gamma = 0.3;
real epsilon =0.1;
real k = 0.3;
real T = 1.0;
real dt = 0.01;
int Ntime = int(T/dt);
// Boundary definitions (region constraint x + y <= 1)
border Gamma1(t=0,1){x=t; y=1-t;}
border Gamma2(t=0,1){x=1-t; y=t;}
border Gamma3(t=0,1){x=t; y=0;}
border Gamma4(t=0,1){x=0; y=t;}
// Mesh generation
int n=10;
mesh Th = buildmesh(Gamma1(n) + Gamma2(n) + Gamma3(n) + Gamma4(n));
plot(Th, wait=true, pdf="mesh.pdf");
// Finite element space
fespace Vh(Th, P1);
Vh f, fold, v;
// Initial conditions
f = (x > k ? x - k : 0);
fold = f;
// Time evolution
for (int n = 0; n < Ntime; n++) {
real t = n * dt;
// Weak form of the partial differential equation (Backward Euler Scheme)
solve P(f, v)
= int2d(Th)(f / dt * v - fold / dt * v)
- int2d(Th)(x^2 * (1 - x - y)^2 * (dx(f) * dx(v)))
+ int2d(Th)((beta * (1 - x - y) - 1+ 2* (1 - x - y) * (1 - x - 2 * y)) * x * (dx(f) * v))
+ int2d(Th)(gamma * x * (dy(f) * v))
- int2d(Th)(epsilon * f * v);
// Update time step
fold[] = f[];
}
// Visualization (3D display)
plot(f, dim=3, wait=true, pdf="solution.pdf");