I have been trying to perform some stability analysis of numerical schemes applied to the linear scalar convection equation on a non-uniform grid. Can we apply the conventional Von Neumann stability analysis for this case? In the case of a uniform grid, we can express the solution in terms of a Fourier series and study the growth of the Fourier modes. Can we write such a Fourier series on a non-uniform grid? I would also like to know of some references on the same subject.

(1) You may transform your equation from non-uniform physical domain to uniform computational domain. (2) you can perform local stability analysis. For steady state solution, you may use the local maximum allowable time step. For time-dependent solutions, you must use the minimum time step of the whole domain. (3) Generaly, stability is a relation between time step and grid size, both can be local or globle. The local stability conditions (Courant number - a dimensionaless number) is independent of grids and depends on schemes only.

The stability analysis can be done in any coordinate system. You can use your Fourier expansion on any grid. However, the grid has to be uniform! If the grid is not uniform, then you should interpolate the solution onto an uniform grid.

When the grid is uniform, a given mode represents a given scale length in your problem. If the grid is not uniform, a given mode of a Fourier expansion can represent several scale lengths at the same time. So it is not clear what it represents. The Fourier expansion is needed to check the exponential growth of the function, and the uniform grid is needed to perform the Fourier transform (discrete transform - or integration) of the function to obtain the amplitude of the modes. You might try to change and transform the coordinate systems into a new one where the grid points are at uniform interval. There your function might look different, but there you can perform the Fourier analysis of the growing unstable modes.

On the stability analysis of convective flow (related to as the Rayleigh Taylors instability) you can check the book of the author who had the same name as you:

Chandrasekhar, Hydrodynamic and Hydromagnetic instability, Dover publication. Cheers,

PG.

I am interested in completely non-uniform grids. So there is no way that I can transform it to a uniform grid. Even if I am able to tranform it to a uniform grid, it is not very clear how the interpolation is to be done. Interpolation may not be good idea especially when the solution is discontinuos. Also, I am not sure that I understand what is meant by a local analysis. How is possible to perform a local Fourier analysis when the very idea is to be able to express the solution as a Fourier series, where the Fourier amplitudes would involve an integration over the whole spatial domain ? Is there no method, other than Fourier analysis, for stability?

Transform back to a uniform grid is irrelevent, because you will lose your original question: stability on a specific mesh grid (which is non-uniform).

Analytical Fourier transform method works for constant coefficent equation. In general, you have to solve an eignevalue problem, and look at the spectrum. Solving a general eigen-equation is not a trivial problem though. The stability is decided by the largest eignevalue in the spectrum, this corresponds to so-called Corant condition in fluid flow. The best stability analysis book I have ever read for fluid flow is 'Spectral methods in Fluid Dynamics' by Canuto etc. Local analysis is more or less pedagogical. You can't expect that solves you problem. However, usually the largest eignevalue in the system is decided by the smallest grid size (inversely proportional). But to excatly determine the criterion, you need to solve the complete eigenvalue problem. For 1d problem, this is not too difficult, you can now solve 1000 order linear eigne value problem easily. But for 2d or 3d, this may pose a serious chanllenge.

By the way, spetral method is much more superior than FD method for stability problem.

Don't give up!

First, read

C. W. Hirt, Heuristic Stability Theory for Finite-Difference Equations, J. Comp. Physics, v. 2 (1968), pp. 339-355. That describes a very useful approximate method that can be extended to more complex differencing schemes with a bit of effort.

Often useful guidelines can be extracted by applying the Fourier analysis locally and using the minimum allowable time step over the entire mesh (a previous poster suggested this as well, and it's used in Hirt's paper above). After all, the Fourier analysis assumes linearity that the NS equations surely don't have!

No matter how you extract your guidelines for selecting a time step, they are only that; guidelines.

Good luck,

Jim

(1). I have used the approach similar to Hirt's method to develop stable methods on non-uniform meshes many years ago. You are basically looking at the properties of the final algebraic equations you are trying to solve. (2). So, Hirt's method is a good starting point. ( It was also published in one of AIAA's special issue on Computational Fluid Dynamics in early 70's)

Hi Praveen,

A different way of doing stability analysis of finite difference schemes is presented in a book by Y.I. Shokin " The method of differential approximation" (springer verlag series in computational physics)

again, there are number of different ways one can do stabilty analysis of difference schemes approximating PDEs 1. spectral method or Von Neumann stability analysis 2. differential approximation method 3. method of frozen coefficients 4. method of energy inequatilities 5. method of separation of variables. 6. methods based on a maximum principle. 7. method of discrete perturbation. etc.. (reference : V.G. Ganzha and E.V. Vorozhtsov " Computer aided analysis of differnce schemes for partial differential equations") Note that performing such analysis "manually" is nearly impossible. There are several computer algebra systems like REDUCE, MACSYMA, MAPLE, MATHEMATICA etc. that are capable of doing this horrible job of book-keeping for you.

The book by Ganzha and Vorozhtsov will provide all the answers about stability on non-orthogonal curvilinear grids also.

Hope this helps

Mayank