A dispersion analysis of uniformly high order, interior and boundaries, mimetic finite difference solutions of wave propagation problems

A preliminary stability and dispersion study for wave propagation problems is developed for mimetic finite difference discretizations. The discretization framework corresponds to the fourth-order staggered-grid Castillo-Grone operators that offer a sextuple of free parameters. The parameter-dependent mimetic stencils allow problem discretization at domain boundaries and at the neighbor grid cells. For arbitrary parameter sets, these boundary and near-boundary mimetic stencils are lateral, and we here draw first steps on the parametric dependency of the stability and dispersion properties of such discretizations. As a reference, our analyses also present results based on Castillo-Grone parameters leading to mimetic operators of minimum bandwidth that have been previously applied in similar physical problems. The most interior parameter-dependent mimetic stencils exhibit a specific Toeplitz-like structure, which reduces to the standard central finite difference formula for staggered differentiation at grid interior. Thus, our results apply to the whole discretization grid. The study done for the 1-D problem could be applied to the discretization of a free surface boundary condition along an orthogonal gridline to this boundary.