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Stefan Bilbao, "Wave and Scattering Methods for the Numerical Integration of Partial Differential Equations"

Wave and Scattering Methods for the Numerical Integration of Partial Differential Equations by Stefan Bilbao Doctoral Dissertation Stanford University 2001

Abstract

Digital filtering structures have recently been applied toward the numerical simulation of distributed physical systems. In particular, they have been used to numerically integrate systems of partial differential equations (PDEs), which are time-dependent, and of hyperbolic type (implying wave-like solutions, with a finite propagation velocity). Two such methods, the multidimensional wave digital filtering and digital waveguide network approaches both rely heavily on the classical theory of electrical networks, and make use of wave variables, which are reflected and transmitted throughout a grid of scattering junctions as a means of simulating the behavior of a given model system. These methods possess many good numerical properties which are carried over from digital filter design; in particular, they are numerically robust in the sense that stability may be maintained even in finite arithmetic. As such, these methods are potentially useful candidates for implementation in special purpose hardware.

In this thesis, the subtext is that such scattering-based methods can and should be treated as finite difference schemes, for purposes of analysis and comparison with standard differencing forms. In many cases, these methods can be shown to be equivalent to well-known differencing approaches--we pay close attention to the relationship between digital waveguide networks and finite difference time domain (FDTD) methods. For this reason, it is probably most useful to think of scattering forms as alternative realizations of these schemes with good numerical properties, in direct analogy with ladder, lattice and orthogonal digital filter realizations of direct form filters. We make use of this correspondence in order to import (from the finite difference setting) two techniques for approaching problems with irregular boundaries, namely coordinate changes, and a means of designing interfaces between grids of different densities and/or geometries. We also make use of the finite difference formulation in order to examine initial and boundary conditions, parasitic modes, and take an extended look at the numerical properties of all the commonly encountered forms of the waveguide network in two and three spatial dimensions.

Another question is of the relationship between wave digital and waveguide network schemes. Although they are quite similar from the standpoint of the programmer, in that the main operation, scattering, is the same in either case, conceptually they are very different. A multidimensional wave digital network is derived from a compact circuit representation of model system of PDEs. The numerical routine is itself a discrete time and space image of the original network. Waveguide meshes, however, are usually formulated as collection of lumped scattering junctions which span the problem domain, connected by bidirectional delay lines. Lacking a multidimensional representation, then, it is not straightforward to design a mesh which numerically solves a given problem. A useful result is that waveguide meshes can be obtained directly from a system by almost exactly the same means as a wave digital network. This unification of the two methods opens the door to a larger class of methods which are of neither type, and yet which consist of the same numerically robust basic building blocks.

On the applied side, special attention is paid to problems in beam, plate and shell dynamics; though these systems are in general much more complex than the transmission line and parallel-plate problems which have been discussed extensively in the literature, they can be dealt with using both wave digital filters and waveguide networks, though several new techniques must be introduced. Several simulations are presented.

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