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

Symmetric hyperbolic systems include many physically relevant systems of PDEs such as Maxwell's equations, the elastic wave equations, and the acoustic equations [L. Ryzhik, G. Papanicolaou, and J. Keller, Wave Motion, 24 (1996), pp. 327--370]. In the current paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with polarized waves, in which the dispersion matrix of the hyperbolic system has eigenvalues with constant multiplicity greater than one over the domain of computation. The new results in this paper include new Gaussian beam equations in the presence of multiple eigenvalues, error estimates for Gaussian beam summation in the symmetric hyperbolic system case, and a new multidirectional Eulerian summation formula which maintains accuracy after the formation of caustics.