Numerical analogs to Fourier and dispersion analysis: development, verification, and application to the shallow water equations

Herein, we present numerical analogs to traditional Fourier and dispersion analyses and validate them with well-characterized phase behavior for classic finite difference and finite element (FE) discretizations of the shallow water equations. Basically, the procedure is to introduce a single wave with known amplitude and phase into the domain, propagate the wave approximately one wavelength using some discretization scheme, and then note its final amplitude and phase. The final state of the wave is then compared with the expected wave form predicted by the continuum equations to determine the propagation behavior of the discretization. After validating the technique, we then examine two case studies: (1) slope limiting schemes within the finite volume framework and (2) lumping coefficients within the selective lumping FE framework. Of the three common slope limiters that we examined, the Superbee limiter has the most promising phase behavior, as it is the least dissipative while maintaining minimal phase error. Using our numerical technique, we were also able to verify the range of values that has been found to be most accurate in practice for the selective lumping coefficient.