[oksero]

Hello all,

I was wondering if someone could explain the basics of the finite difference method, namely higher order 5th or 6th order schemes. And exactly how the solution is solved by an iterative process. Perhaps an example algorithm.

I`m also open to any suggestions for books or lectures that would expand my knowledge in anyway.

[Martin Hegedus]

I've got limited experience, that being said...

This may be a good starting point in regards to expanding knowledge.

High-order accurate, low numerical diffusion methods for aerodynamics, John A. Ekaterinaris.

Also, some hints in regards to FDM,

1) one needs to make sure that the areas, if using generalized coordinates, are compatible with the differencing. If a uniform flow field is assumed, the dirivatives must equal zero.
2) It's been my limited experiences that low dissipation methods are challenging to converge. You might need a dual time stepping procedure.
3) Boundary conditions can be challenging to implement.
4) If you are using artificial dissipation, the implementation may also need to use a higher order approximation.

Good Luck!

[FMDenaro]

The old book of Morton can be a starting point, then many modern books of CFD explain the essence of FDM (Anderson, Pletcher, Hirsch, Peric, etc.).

Speaking in general:

explicit FD: are based on a Lagrange polynomial, therefore higher order derivatives require larger computational stencil

implicit (Compact) FD: are based on the Padè interpolation, you can get high order derivative on compact stencil at the price of solving a linear algebric system.

Each term in the PDE is substituted by some FD formula to get a discrete equation that, depending on the PDE, is solved in direct/iterative way.

[oksero]

Cheers, Thanks for the book recommendations

[Martin Hegedus]

Originally Posted by FMDenaro

The old book of Morton can be a starting point,...

I'm going to confess ignorance. Which book is it that you recommend?

Is it
"Difference Methods for Initial Value Problems" by Richtmyer and Morton, or another?

Thanks.

[FMDenaro]

Originally Posted by Martin Hegedus

I'm going to confess ignorance. Which book is it that you recommend?

Is it
"Difference Methods for Initial Value Problems" by Richtmyer and Morton, or another?

Thanks.

yes,

further, Bill Morton publishede more recently: K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations, An Introduction. Cambridge University Press, 2005