Nonlinear Parabolic BVP: Possible Numerical Methods

[jam_27]

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.

Please also give me a reference with your suggestions so that I can refer it. Please ask me if you need any more information about the PDE.

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

Jam

 

[Clausius2]

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!

 

[jam_27]

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

 

[jam_27]

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

 

[Clausius2]

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

You're welcome. I guess the best edible books for dealing with this stuff are the book of J.D.Anderson "CFD", and Anderson and Tanenhill (CFD and heat transfer).

PS: CFD: computational fluid dynamics.