Absorbing Boundary Conditions For Hyperbolic Systems Matthias

NUMERICAL MATHEMATICS: Theory, Methods and Applications Numer. Math. Theor. Meth. Appl., Vol. xx, No. x, pp. 1-39 (200x)

Absorbing Boundary Conditions for Hyperbolic Systems
Matthias Ehrhardt∗
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany.

Abstract. This paper deals with absorbing boundary conditions for hyperbolic systems in one andtwo space dimensions. We prove the strict well–posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS–stability of the corresponding Lax-Wendroff–type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems. AMSsubject classifications: 65M06, 35L50 Key words: absorbing boundary conditions, hyperbolic system, Engquist and Majda approach, strict well–posedness, GKS–stability.

1. Introduction
This article is concerned with the numerical approximation of hyperbolic partial differential equations that are posed on an unbounded spatial domain (usually N ). When solving this whole space problem numerically oneis facing the problem that one has to confine the computational domain. Typical examples for first order hyperbolic equations are the Maxwell equations, the (linearized) shallow water equations and the classical hydrodynamic equation in semiconductor simulation [29](without heat conduction term). Sometimes it is useful to apply a coordinate transformation with conformal mappings in order totransfer the original whole space problem to a new problem defined on a bounded domain. Unfortunately, the differential equation often becomes quite complicated [34] and this transformation technique of conformal mappings fails, if the solution is oscillating at infinity and turns out to be not suitable for many physical problems [15]. Another strategy is to confine the domain by introducing artificialboundaries without making any changes to the considered differential equation. At these artificial boundaries one defines so–called absorbing boundary conditions (ABCs), which are designed such that the solution of the bounded domain approximates well the solution on the original unbounded domain. The quality of approximation will be higher, if the components leaving
∗

Corresponding author. Emailaddress: ehrhardt@wias-berlin.de (Matthias Ehrhardt) 1 c 200x Global-Science Press

http://www.global-sci.org/nmtma

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Matthias Ehrhardt

the interior of the bounded domain (outflow components) have only small reflections at the artificial boundary. Especially the amplitudes of the waves that are reflected from the artificial boundaries should be as small as possible [9]. There exist a couple ofapplications of absorbing boundary conditions in the literature, e.g. in electro dynamics [5], in fluid dynamics [4] and in geology [6]. In meteorology ABCs are used in local area weather forecasts [8], since the original domain (earth surface) would require a too high computational effort to solve the simulation in the given time frame and coarsening the grid would lead to unsatisfactory results.This work consists of two parts: an analytic part and a numerical (discrete) part. In the first analytic section we will use the technique of pseudodifferential operators [35] to construct a hierarchy of absorbing boundary conditions for linear first order hyperbolic systems. Our procedure closely follows the classical work of Engquist and Majda [9]. Afterwards we will investigate theone–dimensional case and prove that the resulting initial boundary value problems (IBVP) are well-posed in the strict sense of Kreiss and Lorenz [23]. For hyperbolic systems in two spatial dimensions Engquist and Majda already showed that ABCs may give rise to not well–posed problems [9]. In the second, numerical part of this article the derived absorbing boundary conditions are discretized adequately and we...