Advances in High-Order Absorbing Boundaries for Elastodynamics

Abstract eng:
The need for artificial computational boundaries in the solution of exterior wave problems, called “absorbing boundaries” among other names, arises quite often in various fields of application. In solid-earth geophysics they are needed for practically every simulation. Since the mid 90's two classes of methods have emerged as especially powerful: the Perfectly Matched Layer (PML) method and the method of using high-order Absorbing Boundary Conditions (ABCs), which are local and involve no high derivatives. The use of ABCs has been very popular since the early 70's, but the term “highorder ABCs” relates to the ability to implement ABCs of an arbitrarily high order. High-order derivatives are eliminated by introducing auxiliary variables which are discretized on the boundary. Most of the high-order ABCs proposed thus far have been devised for the acoustic (scalar) wave equation. Until recently, the only ABCs proposed for elastic waves were that devised by Tsogka and Joly and that devised by Rabinovich et al. Both turned out to be unstable for long times. In this presentation we report on two recent major advances in the development of high-order ABCs for elastodynamics. First, we have recently devised a new high-order local ABC on an artificial boundary for timedependent elastic waves in unbounded domains, in two dimensions. The stability of this ABC was both proved mathematically, using an energy estimate, and demonstrated numerically. Thus, this is the first known local high-order ABC for elastodynamics which is long-time stable. The elastic medium in the exterior domain is assumed to be homogeneous and isotropic. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The ABC involves a product of firstorder differential operators; all of them are of the Higdon type, except one which is of the LysmerKuhlemeyer type. The initial boundary value problem including this ABC is written as a first-order system, using stresses and velocities as variables. A finite difference scheme in space and time is employed to discretize this system. Numerical experiments demonstrate the performance of the scheme. Second, we have devised a new ABC approach, called the Double Absorbing Boundary (DAB) approach. Instead of discretizing the auxiliary variables only on the boundary, we discretize them in a thin layer adjacent to the boundary, and apply the ABC at the two boundaries which bound this layer. This leads to significant simplification in the numerical scheme, and has some important advantages over the use of a single ABC.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket