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All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be done numerically?

Can you recommend some litterature on this topic?

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Sorry to bring this question back from the grave. I'm assuming you've found your answer by now, but others may be interested.

In general, the time step for finite difference methods (also finite element methods) is limited by the smallest mesh size in your problem. So you carry out your von-Neumann stability analysis as usual, but use the smallest $h$ in your timestep calculation.

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It is possible.

You may want to look at the error dynamics approach:

Sengupta, Tapan K., Anurag Dipankar, and Pierre Sagaut. "Error dynamics: beyond von Neumann analysis." Journal of Computational Physics 226.2 (2007): 1211-1218.

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