# Nonlinear Parabolic BVP: Possible Numerical Methods

## Main Question or Discussion Point

Hi all!

I am coming back here after a long time. Last time I got the answer I was looking for, here. I hope that I will find it again.

I need the possible numerical methods for solving the following PDE: Its a nonlinear parabolic boundary value problem. I want a very stable algorithm as this equation is a part of a bigger code. I don’t want accumulated numerical errors as I am investigating an instability in the bigger code involving a lot of iterations.

The PDE is :
d(N(x,t))/dt=D*d2(N(x,t))/dx2-A*N(x,t)-B*N2(x,t)-C*N3(x,t)-S(x,t)+ R(x)

N(x=a,t)=N(x=b,t)=0 for all t. N(x,t=0)=f(x) . (f(x) is a Gaussian function)

where N2 is N-squared, N3 is N-cube and so on. S(x,t) can be a rapidly oscillating function in x and t. C is very small as compared to the other constants A, B and D (all positive).

Is it possible to use a relaxation method for solving the above PDE? I ask this because the corresponding steady state problem can be solved very effectively using a Relaxation method and I have a efficient code for that already.

Thanks for the help in advance and for having such a great forum.

Jam

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Clausius2
Gold Member
I would solve it with FD, advancing implicitly in time. It is a robust and stable method according to Von Neummann analysis, but your non-linearity may cause numerical instability. Linearize your non linear terms and interate insinde each time t loop for the convergence between the lagged coefficient of the non linear term and the solution itself.

And have fun!

Thanks Clausius2 for the reply. Can you please give me some reference for the technique you suggested.
Jam

Thanks Clausius2 for the reply. Can you please give me some reference for the technique you suggested.
Jam

Clausius2