Nonreflecting Difference Conditions on the Moving and Shape Varying Boundary of the Computational Domain
There is a wide class of unsteady initial boundary-value problems formulated either on the entire Euclidean space R or a large domain D (with boundary conditions), for which the solution u(t, x) needs to be known only on a bounded subdomain of the origina
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1.1 Introduction There is a wide class of unsteady initial boundary-value problems formulated either on the ent ire Euclidean space R or a large domain D (with boundary conditions) , for which the solution u(t, x) needs to be known only on a bounded subdomain of the original domain. Without much loss of generality, we additionally assume that the governing differential equations outside this computational subdomain, as well as boundary conditions at infinity or distant physical boundary, are linear and homogeneous . Let us denote by Din = Din(t) the comput at ional sub domain on which the solution u(t, x) needs to be calculated. This sub domain can change its shape and also move in the course of time. Examples of the corresponding probl ems arise in different physical and engineering applications. Suppose one subjects a homogeneous material to the strong time-dependent stress concentrated in some local area. Then, plasti c deformations or fractures may arise in or near the stress region, while at some distance the deformations are sufficiently small to be described by the Lame syst em of linear elasticity theory. Typically, it is interesting to calculate the solution only on the subdomain, in which large deformations are concentrated. Similarly, when computing external fluid flows around finite configurat ions, one, as a rule , is interested in calculating the solution only in the near field. At the same time, in the far field the flow can be described by the gas dynamics equations linearized around the unperturbed background solution. Other examples are given by the problems of wave diffraction or scattering (acoustic or electromagnetic) . In all these problems, it would be most beneficial from the standpoint of numerical performance to solve the equations only in a relatively small sub domain of interest provided that the influence of the remaining outer part of the domain, along with the boundary conditions at infinity or at a remote physical boundary, is compensated for by the equivalent conditions at an artificial boundary. In this chapter, we consider the problem of how to transfer the boundary conditions from infinity to the artificial boundary of the computational subdomain for abstract systems of time-dependent difference equations. These V. S. Ryaben'kii, Method of Difference Potentials and Its Applications © Springer-Verlag Berlin Heidelberg 2002
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1. Nonreflecting Difference Conditions
equations (systems) are of general type and subject only to the requirements of solvability, uniqueness of the solution, and linearity outside the computational sub domain. Finite-difference or finite-element approximations of many unsteady physical problems belong to this class. In this chapter, we construct a special difference artificial boundary of the computational subdomain so that one can exactly transfer to it boundary conditions from a remote physical boundary or from infinity; we also describe the actual construction of the artificial boundary conditions themselves. In doing this, we follow [83]. Usually,
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