An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - III. Viscoelastic attenuation

Abstract

S U M M A R YWe present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansion in time in which all time derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy-Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and storage space for a desired accuracy can be reduced when applying higher degree approximation polynomials. In addition, we investigate the increase in computing time, when the number of relaxation mechanisms due to the generalized Maxwell body are increased. An application to a well-acknowledged test case and comparisons with analytic and reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-DG approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

A high‐order discontinuous Galerkin method with time‐accurate local time stepping for the Maxwell equations

“…[9,14]). In contrast to this approach Ka¨ser et al proposed in [41] to project the governing equation (19) immediately onto the space spanned by the DG basis functions. They obtain a system of ordinary differential equations, which describes the temporal evolution of the expansion coefficients U ðmÞ l ðtÞ.…”

Optimized viscoelastic wave propagation for weakly dissipative media

“…It is common practice (Emmerich & Korn 1987;Blanch et al 1995;Komatitsch & Tromp 2002b;Graves & Day 2003;Kristek & Moczo 2003;Käser et al 2007;Savage et al 2010) to find the medium parametrization a j , ω j by choosing the relaxation frequencies ω j a priori, mostly logarithmically spaced in the frequency range of interest. Then, the a j can be found by sampling Q(ω) at a finite number of frequencies ω k and solving an overdetermined inverse problem.…”

“…(5), over time to determine the values of the memory variables at the next time step: for example, finite differences (Day & Minster 1984), some unspecified second-order scheme (Emmerich & Korn 1987), second-order central difference (Kristek & Moczo 2003), ADER (Käser et al 2007) or fourth-order Runge-Kutta (Komatitsch & Tromp 2002a;Savage et al 2010). To our best knowledge, integrating the equation analytically has not been published in the seismological community.…”

“…These methods are still common today (e.g. Komatitsch & Tromp 2002b;Graves & Day 2003;Kristek & Moczo 2003;Käser et al 2007;Fichtner et al 2009;Savage et al 2010), a more complete summary and historical overview is given by Carcione (2007) and Moczo et al (2014). In this study, we suggest several improvements to this scheme leading to better accuracy in the medium representation at zero extra cost.…”

“…Blanch et al 1995;Käser et al 2007). Day (1998) suggested a method called the coarsegrained memory variable approach to redistribute the medium properties on the subwavelength scale such that it behaves the same for the wavefield, but is computationally significantly less expensive.…”