Nonlinear Parabolic BVP: Possible Numerical Methods

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Discussion Overview

The discussion revolves around potential numerical methods for solving a nonlinear parabolic boundary value problem (BVP) represented by a partial differential equation (PDE). The focus is on finding a stable algorithm to minimize numerical errors, particularly in the context of a larger code that investigates instabilities over multiple iterations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks numerical methods for a specific nonlinear parabolic PDE, emphasizing the need for stability due to its role in a larger code.
  • Another participant suggests using finite difference (FD) methods with implicit time advancement, noting its robustness and stability but warns that non-linearity may introduce numerical instability.
  • This participant recommends linearizing the non-linear terms and iterating within each time loop to achieve convergence.
  • There are requests for references to support the suggested techniques, indicating a desire for further reading on the methods proposed.
  • A later reply provides references to specific texts on computational fluid dynamics (CFD) that may be relevant to the discussion.

Areas of Agreement / Disagreement

The discussion does not reach a consensus on the best numerical method, as participants present different approaches and techniques without resolving which is superior or more appropriate for the problem at hand.

Contextual Notes

The discussion highlights the complexity of the PDE, including the presence of non-linear terms and boundary conditions, which may affect the choice of numerical methods. The specific characteristics of the oscillating function S(x,t) and the smallness of constant C compared to others are also noted but not fully explored.

Who May Find This Useful

Researchers and practitioners working on numerical methods for solving nonlinear parabolic PDEs, particularly in the fields of computational fluid dynamics and applied mathematics, may find this discussion relevant.

jam_27
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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
 
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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
 
jam_27 said:
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.
 

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