spiral_pde

spiral_pde, a MATLAB code which solves a pair of reaction-diffusion partial differential equations (PDE) over a rectangular domain with periodic boundary condition, whose solution is known to evolve into a pair of spirals.

Because of the large number of variables, ode45() might not be a suitable solver, since it can run out of memory. For this problem, a simple Forward Euler ODE approach is used instead.

The PDE has the form:

        du/dt =         del u + u * ( 1 - u ) * ( u - ( v + beta ) / alpha ) ) / epsilon
        dv/dt = delta * del v + u - v
      

The domain is the square 0 <= x <= 80, 0 <= y <= 80, with zero Neumann boundary conditions.

The initial conditions are

        u = 0       for x < 40
            1       for     40 < x
        v = 0       for y < 40
            alpha/2 for     40 < y
      

The parameters are

        alpha   = 0.25
        beta    = 0.001
        delta   = 0.0
        epsilon = 0.002
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Reference:

Willem Hundsdorfer, Jan Verwer,
Numerical solution of time dependent advection-diffusion-reaction equations,
Springer Series in Computational Mathematics,
Volume 33, Berlin, 2003.

Source Code:

• laplacian9_torus.m uses a 9 point Laplacian stencil on a rectangular periodic grid.
• spiral_deriv.m evaluates the right hand side of the system of ODEs.
• spiral_initial_condition.m sets the initial values of the solution.
• spiral_parameters.m returns parameters for the spiral PDE.

Tests:

• spiral_pde_test.m calls all the tests.
• spiral_pde_test.sh runs all the tests.
• spiral_pde_test.txt the output file.

Plots were made every 500 steps.

• spiral_0001.png
• spiral_0501.png
• spiral_1001.png
• spiral_1501.png
• spiral_2001.png
• spiral_2501.png
• spiral_3001.png
• spiral_3501.png
• spiral_4001.png
• spiral_4501.png
• spiral_5001.png