Compact Finite Difference Schemes for the Acoustic Wave Equation

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In this work, we implement and compare two fourth-order compact finite difference (CFD) methods to model the propagation of acoustic waves on a 1-D domain. The first scheme is based on standard implicit CFD constructed on nodal grids and must solve a couple of tridiagonal linear systems at each time iteration. The formulation of the second method uses the novel staggered mimetic CFD that explicitly approximate derivatives at cell center points. Mimetic FD operators inherit this name from the fact that some conservation properties fulfilled by the continuous divergence, gradient, and curl operators, are also satisfied by the discrete mimetic counterparts. Both CFD methods are combined to high-order Runge-Kutta schemes for time integration. In our current preliminary experiments, results show that the mimetic CFD scheme is slightly more accurate and shows similar fourth-order convergence than the nodal scheme. In this paper, we also compare the simulation times spent by the compact schemes to an explicit Leap-frog staggered solver. This comparison reflects the high computational efficiency of the mimetic CFD scheme, and along with its accurate results, suggests the potential of this method for multidimensional wave propagation problems.