An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms

Abstract

SUMMARY We present a new numerical approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high‐order accuracy in space and time on unstructured triangular meshes. We combine a discontinuous Galerkin (DG) method with the ideas of the ADER time integration approach using Arbitrary high‐order DERivatives. The time integration is performed via the so‐called Cauchy‐Kovalewski procedure using repeatedly the governing partial differential equation itself. In contrast to classical finite element methods we allow for discontinuities of the piecewise polynomial approximation of the solution at element interfaces. This way, we can use the well‐established theory of fluxes across element interfaces based on the solution of Riemann problems as developed in the finite volume framework. In particular, we replace time derivatives in the Taylor expansion of the time integration procedure by space derivatives to obtain a numerical scheme of the same high order in space and time using only one single explicit step to evolve the solution from one time level to another. The method is specially suited for linear hyperbolic systems such as the heterogeneous elastic wave equations and allows an efficient implementation. We consider continuous sources in space and time and point sources characterized by a Delta distribution in space and some continuous source time function. Hereby, the method is able to deal with point sources at any position in the computational domain that does not necessarily need to coincide with a mesh point. Interpolation is automatically performed by evaluation of test functions at the source locations. The convergence analysis demonstrates that very high accuracy is retained even on strongly irregular meshes and by increasing the order of the ADER–DG schemes computational time and storage space can be reduced remarkably. Applications of the proposed method to Lamb's Problem, a problem of strong material heterogeneities and to an example of global seismic wave propagation finally confirm its accuracy, robustness and high flexibility.

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

“…However, there are many alternatives including the space-time DG method [33,37] and the ADER approach using arbitrary high-order derivatives [38,39,40]. For example, the ADER scheme with the DG method has been successfully applied to model elastic, poroelastic, and electromagnetic wave propagation, see for example [30,32,15,41]. Reviews of developments in the DG methods in are given in [42,34].…”

“…A broad ranging review of computational poroelasticity is given 30 in [12]. We refer also to the recent papers [13,19,20] who work in a finite volume setting.…”

“…Since then the method has been extensively analysed [22,23,24,25,26,27]. Furthermore, the DG method and many variants have been applied to many problems, including first-and second-order hyperbolic problems such as Maxwell's equations and the elastic and acoustic wave equations [28,29,30, 31, 40 …”

Higher-order Source-wavefield Reconstruction for Reverse-time Migration from Stored Values in a Boundary Strip Just One Point Wide

“…Examples relevant to seismic modeling include higher-order time stepping (Dablain, 1986;Käser and Dumbser, 2006) and the modified equation approach (Shubin and Bell, 1987). This paper explores the stability and accuracy of our scheme in an experimental way.…”

On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids

“…Various discretization methods have been proposed for this task, e.g., Virieux (1984Virieux ( , 1986 with finite difference methods and Komatitsch & Vilotte (1998); Käser & Dumbser (2006) with finite element methods. Although the finite element approach offers more flexibility in dealing with complicated geometry and local refinement, the finite difference approach is still notably popular and widely employed in the seismic exploration community due to its efficiency.…”