Spectral Approach to the Construction of Nonreflecting Boundary Conditions
We consider a family of time-dependent problems with solutions u(x, t) defined for t ≥ 0 in a domain D consisting of the whole x-plane excluding possibly a finite number of subdomains different for each problem, but lying inside the disk $$ S_R = \left\{
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Introduction We consider a family of time-dependent problems with solutions u(x, t) defined for t ;::: 0 in a domain D consisting of the whole x-plane excluding possibly a finite number of subdomains different for each problem, but lying inside the disk for R < 1. We assume that equations for each problem coincide and are linear, homogeneous and do not depend explicitly on t outside the disk 8R with R = const. It is also assumed that ult=o = ~~ It=o = .. . = 0 and that each problem of the family has a unique solution. Suppose that the solution of each problem is to be calculated only in the part of its domain that lies inside 8 1 , t ;::: O. The problem is treated in the finite-difference formulation; we make use the accurate nonreflecting artificial boundary conditions (NRABCs) constructed in the general case in [83] . For the case of a half-space (see also Sect. [179]), note that the direct employm ent of these conditions, generally speaking, is not more efficient than the procedure consisting of the direct solution of the initial finite-difference problem. We propose and discuss here some preliminary mathematical tools which allow us to reduce computational costs and provide a sufficiently accurate approximation of NRABCs for any problem from a given family. It is for the first time here that the spectral approach, in conjunction with the method of difference potentials, is implemented for the nonstationary problems of this type. The spectral approach and the const ruction of mathematical tools (the "framework" of NRABCs), developed and illustrated here for a simple test case, can be generalized to a multidimensional case x = (Xl, X2 ,' •• , X n ), n ;::: 2, t > 0, and an arbitrarily shaped artificial boundary of the computational sub domain for problems with coefficients depending on t. For example, one can set n = 3 and replace x~ + x~ = 1 by the boundary of a tube xi + x~ ::; 1, Ixal < L, where 2L is tube length. The medium surrounding the tube may be described by differential equations with variable coefficients. V. S. Ryaben'kii, Method of Difference Potentials and Its Applications © Springer-Verlag Berlin Heidelberg 2002
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2. Spectral Approach to the Construction of NRABCs
Spectral methods based upon the use of special basis functions have been widely used recently for the numerical solution of problems in bounded domains [176]-[178]. Applications of these methods to the construction of artificial boundary conditions are unknown to the authors. The treatment in this chapter follows, in the main, the treatment in [84J .
2.1 Finite-Difference Nonreflecting Boundary Conditions Conditions on the artificial boundary of the computational subdomain are called nonrefiecting or accurate if they equivalently replace finite-difference equations outside the artificial boundary separating the bounded computational subdomain from the whole, possibly unbounded, domain of the initial nonstationary problem. Examples of NRABCs were constructed in Sect . 2.1 for problems with constant coefficients in the half-space an
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