A stable auxiliary differential equation perfectly matched layer condition combined with low-dispersive symplectic methods for solving second-order elastic wave equations

Abstract

The stable implementation of the perfectly matched layer (PML), one of the most effective and popular artificial boundary conditions, has attracted much attention these years. As a type of low-dispersive and symplectic method for solving seismic wave equations, the nearly-analytic symplectic partitioned Runge-Kutta (NSPRK) method has been combined with split-field PML (SPML) and convolutional complex-frequency shifted PML (C-CFS-PML) previously to model acoustic and short-time elastic wave modelings, not yet successfully applied to long-time elastic wave propagation. In order to broaden the application of NSPRK and more general symplectic methods for second-order seismic models, we formulate an auxiliary differential equation (ADE)-CFS-PML with a stabilizing grid compression parameter. This includes deriving the ADE-CFS-PML equations and formulating an adequate time integrator to properly embed their numerical discretizations in the main symplectic numerical methods. The resulting (N)SPRK+ADE-CFS-PML algorithm can help break through the constraint of at most second-order temporal accuracy that used to be imposed on SPML and C-CFS-PML. Especially for NSPRK, we implement the strategy of neglecting the treatment of third-order spatial derivatives in the PML domain and obtain an efficient absorption effect. Related acoustic and elastic wave simulations illustrate the enhanced numerical accuracy of our ADE-CFS-PML compared with SPML and C-CFS-PML. The elastic wave simulation in a homogeneous isotropic medium shows that compared to NSPRK+C-CFS-PML, the NSPRK+ADE-CFS-PML is numerically stable throughout a simulation time of 2 s. The synthetic seismograms of the 2D acoustic SEG salt model and the two-layer elastic model demonstrate the effectiveness of NSPRK+ADE-CFS-PML for complex elastic models. The stabilization effect of the grid compression parameter is verified in the final homogeneous isotropic elastic model with free-surface boundary.