Compact finite difference modeling of 2-D acoustic wave propagation

Abstract

We present two fourth-order compact finite difference (CFD) discretizations of the velocity–pressure formulation of the acoustic wave equation in 2-D rectangular grids. The first method uses standard implicit CFD on nodal meshes and requires solving tridiagonal linear systems along each grid line, while the second scheme employs a novel set of mimetic CFD operators for explicit differentiation on staggered grids. Both schemes share a Crank–Nicolson time integration decoupled by the Peaceman–Rachford splitting technique to update discrete fields by alternating the coordinate direction of CFD differentiation (ADI-like iterations). For comparison purposes, we also implement a spatially fourth-order FD scheme using non compact staggered mimetic operators in combination to second-order leap-frog time discretization. We apply these three schemes to model acoustic motion under homogeneous boundary conditions and compare their experimental convergence and execution times, as grid is successively refined. Both CFD schemes show four-order convergence, with a slight superiority of the mimetic version, that leads to more accurate results on fine grids. Conversely, the mimetic leap-frog method only achieves quadratic convergence and shows similar accuracy to CFD results exclusively on coarse grids. We finally observe that computation times of nodal CFD simulations are between four and five times higher than those spent by the mimetic CFD scheme with similar grid size. This significant performance difference is attributed to solving those embedded linear systems inherent to implicit CFD.