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simulkade / FVTool

Finite volume toolbox for Matlab/Octave
http://fvt.simulkade.com
BSD 2-Clause "Simplified" License
99 stars 56 forks source link

solving reaction-diffusion(-advection) equations? #14

Closed janelias1 closed 5 years ago

janelias1 commented 7 years ago

Hello, Many thanks for this great piece of code. I am just wondering if it could be used also for reaction diffusion problems? like for the Fisher-KPP equation u_t = Laplacian(phi(u)) + u(1-u) where phi(u)=u^m. While I can solve u_t = Laplacian(phi(u)) with FVTool, I don't know how to treat the nonlinear reaction term. How about systems of nonlinear PDEs? Thanks a lot! Cheers, Jan

simulkade commented 7 years ago

Hi @janelias1

Yes , you can solve this equation by first linearizing the source term u(1-u) = u-u^2. The first u is discretized as M_s1 = linearSourceTerm(createCellVariablem, 1.0)) The second term is linearized to

u^2 = u_0^2 + 2u_0(u-u_0) = 2u_0*u - u_0^2

which can be linearized as

RHS_s1 = constantSourceTerm(u_val.^2) % u_val is a cell variable containing the current value of u
M_s2 = linearSourceTerm(2*u_val)

Send me your code if it still does not work.

janelias1 commented 7 years ago

Thanks a lot! I tried indeed to use this linearisation (Taylor expansion) before but with no success: I don't obtain any travelling wave solution of the Fisher-KPP equation. So I thought that the problem is with the reaction term. Can you please have a look at the code? I have just sent the code to your outlook.com email. Many thanks in advance.