Overview of computation strategies on the dispersion analysis for explicit finite difference solution of acoustic wave equation

Finite-difference(FD)method is the most extensively employed numerical modeling technique.Nevertheless,when using the FD method to simulate the seismic wave propagation,the large spatial or temporal sampling interval can lead to dispersion errors and numerical instability.In the FD scheme,the key factor in determining both dispersion errors and stability is the selection of the FD weights.Thus,How to obtain appropriate FD weights to guarantee a stable numerical modeling process with minimum dispersion error is critical.The FD weights computation strategies can be classified into three types based on different computational ideologies,window function strategy,optimization strategy,and Taylor expansion strategy.In this paper,we provide a comprehensive overview of these three strategies by presenting their fundamental theories.We conduct a set of comparative analyses of their strengths and weaknesses through various analysis tests and numerical modelings.According to these comparisons,we provide two potential research directions of this field:Firstly,the development of a computational strategy for FD weights that enhances stability;Secondly,obtaining FD weights that exhibit a wide bandwidth while minimizing dispersion errors.

Nodal Discontinuous Galerkin Method for Aeroacoustics and Comparison with Finite Difference Schemes

A nodal discontinuous Galerkin formulation based on Lagrange polynomials basis is used to simulate the acoustic wave propagation. Its dispersion and dissipation properties for the advection equation are investigated by utilizing an eigenvalue analysis. Two test problems of wave propagation with initial disturbance consisting of a Gaussian profile or rectangular pulse are performed. And the performance of the schemes in short,intermediate,and long waves is evaluated. Moreover,numerical results between the nodal discontinuous Galerkin method and finite difference type schemes are compared,which indicate that the numerical solution obtained using nodal discontinuous Galerkin method with a pure central flux has obviously high frequency oscillations for initial disturbance consisting of a rectangular pulse,which is the same as those obtained using finite difference type schemes without artificial selective damping. When an upwind flux is adopted,spurious waves are eliminated effectively except for the location of discontinuities. When a limiter is used,the spurious short waves are almost completely removed. Therefore,the quality of the computed solution has improved.

Optimization of the MUSCL scheme by dispersion and dissipation

A second-order optimized monotonicity-preserving MUSCL scheme(OMUSCL2) is developed based on the dispersion and dissipation optimization and monotonicity-preserving technique.The new scheme(OMUSCL2) is simple in expression and is easy for use in CFD codes.Compared with the original second-order or third-order MUSCL scheme,the new scheme shows nearly the same CPU cost and higher resolution to shockwaves and small-scale waves.This new scheme has been tested through a set of one-dimensional and two-dimensional tests,including the Shu-Osher problem,the Sod problem,the Lax problem,the two-dimensional double Mach reflection and the RAE2822 transonic airfoil test.All numerical tests show that,compared with the original MUSCL schemes,the new scheme causes fewer dispersion and dissipation errors and produces higher resolution.