Comparison of time and frequency domain approaches to simulate propagation of weak shocks.

Abstract

Simulations of acoustic fields using finite-difference methods are performed either in time or frequency domains. A method of fractional steps with an operator splitting procedure is frequently applied to solve nonlinear equations of the evolution type. Either time or frequency domain solvers can be used to calculate different terms in the equation over a propagation grid step. In this work, several algorithms that have been used to simulate quadratic nonlinear term in the Burgers or KZK-type evolution equations are applied to model the propagation of weak shocks. Shock capturing schemes of Godunov type, exact analytic solution with further extrapolation of the waveform over a uniform temporal grid, time-domain conservative schemes, direct modeling in the frequency domain, and asymptotic spectral approach are compared. The parameters of the schemes that would provide the results of the same accuracy, an artificial absorption necessary for stability of the schemes, resolution of shocks, and internal viscosity of the algorithms are discussed. It is shown that the Godunov-type algorithm is better suited to model weak shocks with sufficient accuracy achieved with only three temporal grid points per shock. [Work supported by the RFBR and NIH EB007643 grants.]