Wave Propagation In Staggered-Grid Finite-Difference Models With Boundaries

Finite-difference numerical models are widely used in acoustics, electrodynamics and fluid dynamics. In particular, the so-called C-grid (or Yee grid) is a popular staggered-grid formulation, with excellent conservation properties and a natural positioning of nodes. However, domain boundaries are typically treated as staircases, and these degrade the accuracy of the numerical solutions. Here that degradation is quantified for various linear wave propagation problems in idealised geometries. Here, the discrete solution is studied for three important models: (i)wave propagation along a channel, (ii)wave reflection at a planar wall, (iii)the long-time dynamics of waves sloshing in two simple closed domains (square and circle). The first two problems are solved analytically, using asymptotics to examine the limit of small grid spacing h, with expressions for the wavespeed reduction (in (i)) and a phase error (in (ii)) being derived. The third problem is examined numerically, using a high-order time-stepping scheme so that the effects of the staircase boundaries can be isolated. We typically find first-order convergence in grid spacing h, although there are some variations, according to whether we consider convergence in velocities or pressure, and also whether we use L_2 or L_infinity-norm. Some extensions to the propagation of internal waves in a density stratified medium are also considered, which is a less standard scenario, but which has considerable significance in geophysical fluid dynamics