Computing high frequency solutions of symmetric hyperbolic systems with polarized waves

Abstract

We develop computational methods for high frequency solutions of general symmetric hyperbolic systems with eigenvalue degeneracies (multiple eigenvalues with constant multiplicities) in the dispersion matrices that correspond to polarized waves. Physical examples of such systems include the three dimensional elastic waves and Maxwell equations. The computational methods are based on solving a coupled system of inhomogeneous Liouville equations which is the high frequency limit of the underlying hyperbolic systems by using the Wigner transform [15]. We first extend the level set methods developed in [6] for the homogeneous Liouville equation to the coupled inhomogeneous system, and find an efficient simplification in one space dimension for the Eulerian formulation which reduces the computational cost of two-dimesnional phase space Liouville equations into that of two one-dimensional equations. For the Lagrangian formulation, we introduce a geometric method which allows a significant simplification in the numerical evaluation of the energy density and flux. Numerical examples are presented in both one and two space dimensions to demonstrate the validity of the methods in the high frequency regime.

On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model

“…The behavior of high frequency waves are usually considered in numerical research for its different phenomena which lower frequency waves don't have. For example, [17] developed numerical methods for high frequency solutions of general symmetric hyperbolic systems, and [18] focuses on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD). Both of them use a WKB kind initial data, i.e.…”

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

“…As a consequence, the eigenvectors are independent of p. With the help of the following simple theorem, we can make use of these facts to reduce the system (3.5) to one-dimensional computations. This trick was first used in [10].…”

“…The second difficulty was overcome simply by making a careful choice and, by trial and error, showing that our chosen anzatz gives meaningful results. To overcome the last difficulty, we derive a form for our coupling matrix which matches the one discovered in [26] thereby showing a deep connection between this new Gaussian beam method and our previous work [10].…”

“…The limiting equation is the Liouville equation which does not depend on ε, permitting large time steps and mesh sizes. In a previous paper [10], we developed a numerical method based on this approach for the problem under study. One should note that the Liouville equation based classical or geometric optic limit approach, derived via the limit ε → 0, does not offer a good accuracy near the caustics.…”