Coxeter Lecture Series
March 11, 12, 14, 2002 at 3:30 pm
Reception will be held on March 11th, after the Lecture.
Randall J. LeVeque
Applied Mathematics Department, University of Washington
Solving Wave Propagation Problems in Heterogeneous Media
Audio of Lectures - Talk
1 - Talk
2 - Talk
3
Photos of the Event
Abstract:
Wave propagation problems typically lead to hyperbolic systems
of partial differential equations, with coefficients determined
by the material properties of the medium through which the wave
propagates. These coefficients determine the propagation speed
and characteristic structure of the waves. Many practical problems
require modeling the propagation of waves through heterogeneous
media, in which the coefficients are spatially varying and often
discontinuous across sharp interfaces between different materials.
In these three lectures I will present an overview of some wave
propagation problems and a class of numerical methods that can
be used for their solution.
In the first lecture I will introduce the idea of a generalized
Riemann problem at an interface between two materials and show
how the wave structure of this Riemann solution relates to classical
reflection and transmission coefficients. This viewpoint gives
some insights into homogenization theory of waves in periodic
media. This will be illustrated for an acoustics problem in one
dimension. Nonlinear elastic wave propagation in a periodic medium
will also be discussed, where the dispersive behavior caused by
the periodic structure can lead to the appearance of solitary
waves.
In the second lecture I will discuss high-resolution numerical
methods based on the Riemann solution. These finite volume Godunov-type
methods, originally developed for the accurate calculation of
shock waves, are also useful for solving wave propagation problems
in heterogeneous media. Nonlinear problems with spatially-varying
flux functions can also be solved, including nonlinear elastic
wave propagation problems.
The third lecture will concern multidimensional problems and
numerical methods for their solution. The focus will be on acoustics
and linear elastodynamic problems, though the methods can also
be extended to multidimensional nonlinear problems.
|
