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

Abstract

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.

General superpositions of Gaussian beams and propagation errors

“…Our starting point is [9], and we refer the reader to it for more references to earlier results on superpositions of beams. In addition, some recent effort has also been made to extend the Gaussian beam method to more complex settings such as symmetric hyperbolic systems with polarized waves [3], the Schrödinger equation with discontinuous potentials [4], and wave equations in bounded convex domains [1,2].…”

“…We have ∇u GB (x, 0) = −(k(1 + k)x(1) , k(1 + k) −1 x (2) , kx(3) )u GB (x, 0). This givesk −m ||∇u GB (·, 0)|| 2 L 2 = c 1 k 2 (1 + k) 2+r−m (k(1 + k)) −1−r/2 (k/(1 + k)) −(m−r)/2 k −(d−m)/2 + c 2 k 2 (1 + k) −2+r−m (k(1 + k)) −r/2 (k/(1 + k)) −1−(m−r)/2 k −(d−m)/c 3 k 2 (1 + k) r−m (k(1 + k)) −r/2 (k/(1 + k)) −(m−r)/2 k −1−(d−m)/c 1 k 1−d/2 (1 + k) 1−m/2 + c 2 k 1−d/2 (1 + k) −1−m/2 + c 3 k 1−d/2 (1 + k) −m/2 .…”