An analysis of 1D finite-volume methods for geophysical problems on refined grids

This paper examines high-order unstaggered symmetric and upwind finite-volume discretizations of the advection equation in the presence of an abrupt discontinuity in grid resolution. An approach for characterizing the initial amplitude of a parasitic mode as well as its decay rate away from a grid resolution discontinuity is presented. Using a combination of numerical analysis and empirical studies it is shown that spurious parasitic modes, which are artificially generated by the resolution discontinuity, are mostly undamped by symmetric finite-volume schemes but are quickly removed by upwind and semi-Lagrangian integrated mass (SLIM) schemes. Slope/curvature limiting is insufficient to completely remove these modes, especially at low forcing frequencies where the incident wave can act as a carrier of the parasitic mode. Increasing the order of accuracy of the reconstruction at the grid interface is effective at removing noise from the lowest-frequency incident modes, but insufficient at high frequencies. It is shown that this analysis can be extended to the 1D linear shallow-water equations via Riemann invariants.