Multiscale Analysis of Hyperbolic Partial Differential Equations

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The research supported by this award concerns the qualitative behavior of solutions of linear and nonlinear hyperbolic partial differential equations. Such equations describe wave propagation in situations when signals travel at finite speed, e.g. acoustics, electromagnetism, compressible fluids, ... etc. The first set of problems concerns nonlinear internal layers. Such layers represent waves which have wave fronts occupying a very thin region about a smooth surface in space time. The wave front separates regions where the values of the solution are quite different. The examples of propagating flame fronts and reaction fronts give the idea. The mathematical results will lead to algorithms for approximations whose error tends to zero as the width decreases. The problem proposed is the behavior of planar layers at times which are large relative to the reciprocal of the width. One expects to see diffractive effects and there are virtually no known mathematical results for this type for layers. It is likely that there will be analogous results for weakly dissipative boundary layers. The second problems concern Berenger?s perfectly matched layer algorithms for the computation of wave propagation in unbounded domains. Careful analysis indicates that the layers are not perfectly matched. The detailed analysis of this phenomenon and its consequences is important given the wide use of these methods. The third family of problems concern situations where stability estimates are derived with the aid of pseudodifferential operators. Such operators spread supports which makes them seemingly inappropriate for the study of supports. The project proposes the study of sharp finite speed and uniqueness of the Cauchy problem at space like hypersurfaces for symmetrizable hyperbolic systems. Finally the propagation of short wavelength waves through perturbed periodic media will be studied when the period and wavelength are of comparable size. The interest is in propagation for long distances on which diffractive effects are expected to take place. This research project addresses questions that arise in mathematical models for wave propagation phenomena, i.e. physical situations in which signals can travel at finite speed and interact with one another and with the medium that they traverse. Such signals can take the form of wave fronts (such as the spherical light flash from a camera, sound waves made by clapping hands, or a flame front in a combustion engine) or of rays (as in optics), among others. There is a common class of mathematical models for such phenomena that has been employed very successfully in applications as diverse as chemical engineering, the design and use of radar equipment and ultrasound transceivers, computer graphics, and fiber optics. There are four parts to this project. The first will add to the understanding of models for wave fronts that interact. The second part will examine a very widely used computational method for wave phenomena in unbounded spatial regions (e.g. radar signals bouncing off a plane) and try to explain some anomalies that make this method less reliable than is generally assumed. In the third part, the proposer will establish that a wide class of models does indeed exhibit the finite signal speed in all possible situations. Finally, the fourth project aims at a better understanding of light signals that travel through photonic materials over long distances. The PI will continue his educational efforts through lecture series, published notes, counseling, mentoring of thesis students, and close contact with laboratories in engineering schools and industry.