Optimal Finite Difference Schemes for the Helmholtz Equation with PML

Abstract:

An efficient and accurate numerical scheme for solving the seismic wave equations is a key part in seismic wave propagation modeling. The pollution effect of high wavenumbers (the accuracy of the numerical results often deteriorates as the wavenumber increases) plays a critical role in the accuracy of these numerical schemes and it is inevitable in two and three dimensional Helmholtz equations. Optimal finite difference methods can offer a remedy to this problem; however, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. This study develops a number of optimal finite difference schemes for solving the Helmholtz equation in the presence of PML. In doing so, we implement two common strategies, derivative-weighting and point-weighting strategies, for constructing these schemes. Furthermore, a challenge for developing such methods is being consistent with the Helmholtz equation with PML. Thus, analytical and numerical proofs are provided to show the consistency of the schemes. Moreover, for each developed optimal finite difference method, error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, some optimal parameters strategies for each optimal finite difference schemes are recommended. Furthermore, several examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.

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