Total variation diminishing scheme for advection or advection dispersion equation
 

HI I want to write a total variation diminishing code for the simple advection equation and advection diffusion transport equation. I am using excel VBA to write a code. I have use bacward difference in time and central difference implicit scheme for space. The scheme is showing a lot of oscillation so I want to use the total variation diminishing scheme. Does anyone have a sample code with discretization which I can look at and start developing code ? Thank you so much.

Dont use a central scheme for your advection, use upwind, and your oscillations will go away!

Thank you for your reply.

I think the implicit scheme is stable but I want to know how to implement the TVD scheme for diffusion advection equation. I knew some theory about the TVD scheme but I am having difficulty on how to start the scheme. Can anyone suggest me any sources or provide sample code to start with ? I am working on this code as a part of my research and one paper has stated that using TVD scheme gives better result. So, I want to see how does the result vary when we use TVD scheme.

Thanks.

Hi Cooljd,

check this book:
http://www.amazon.com/Introduction-C.../dp/0131274988

You will learn everything you have ever wanted to know about TVD implementation. very usefull wand clear.

Hi Leflix,

Thank you for your reply. As the name of the book suggests, it will talk more about the finite volume method. I am implementing finite difference method and I am completely new to finite volume method. Do you still suggest to go and read that book or is there any specific book for the implementation of TVD scheme in Finite difference method ?

Thanks a lot.

Yes you are right this book is more devoted to finite volume, but you must know that finite volume implementation is strictly equivalent to finite difference one in case of orthogonal regular grids. It will be easy for you to "translate" from finite volume indicated in the book to what you know in finite difference. Once you have understood that in finite volume phi_(e) corresponds to phi_(i+1/2) in finite difference the job is done !!

The main task is to understand how upwind-biased TVD schemes are implemented regardless the type of discretization method.

I guess there exist some books about TVD schemes using finite difference method. But if I could give you an advise, if you are not specifically interested in high-order schemes I would advise you to switch from finite difference to finite volume for CFD problems.

Please check the books 'high resolution methods for incompressible and low speed flows' and 'Riemann Solvers and Numerical Methods for Fluid Dynamics'. The second one was more popular but I like the first one, you can find some fortran codes in the first book.

Any first order scheme is a TVD scheme. Use upwind discretization. As Leflix mentioned discretized equation of finite volume method and finite difference method looks alike on a uniform grid spacing.

The following books have some direct formulation of finite difference methods.
Computational Fluid Mechanics and Heat Transfer By John C. Tannehill, Dale Arden Anderson, Richard H. Pletcher

Computational Fluid Dynamics By T. J. Chung