In PDEs, it is typical to find out how dissipative or dispersive a numerical method is by writing down the modified PDE corresponding to the numerical method, and seeing if that modified PDE contains dispersive or dissipative terms.
However, I'm dealing with a numerical method for which it seems incredibly hard to write down the modified PDE (due to the basis functions used for the spatial discretization).
Is there some acceptable way of measuring numerical dissipation and dispersion in my numerical method when solving a purely hyperbolic PDE?
I thought about Fourier transforming the numerical solution and looking at the Fourier coefficients. Unfortunately, the PDE is on a manifold, and I'm not sure the Fourier approach would work in this scenario.
Thanks!
If anyone's interested, I found an answer (Professor Luca Bonaventura answered via ResearchGate). There are indeed fairly common measures for numerical dissipation and dispersion; these are statistical measures. They are outlined in the following reference, section 6.
"A Two-Step Scheme for the Advection Equation with Minimized Dissipation and Dispersion Errors", Lawrence L. Takacs. http://journals.ametsoc.org/doi/abs/10.1175/1520-0493%281985%29113%3C1050%3AATSSFT%3E2.0.CO%3B2