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.

Symmetric hyperbolic systems include many physically relevant systems of partial differential equations (PDEs) such as Maxwell's equations, the elastic wave equations and the acoustic equations [26]. In this paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with constant degeneracy that corresponds to polarized waves, in which the dispersion matrix of the hyperbolic system has eigenvalues with constant degeneracy over the domain of computation. The new results in this paper include new Gaussian beam equations in the presence of eigenvalue degeneracy, improved error estimates for Gaussian beam summation and a new multi-directional Eulerian summation formula which maintains accuracy after the formation of caustics.