Global geometrical optics method

We develop a novel approach, named the global geometrical optics method, for the numerical solution to wave equations in the high-frequency regime. The initial Cauchy data is assumed to be in the WKB form. We first study the Schrödinger equation, and then extend relevant results to the general scalar wave equations. The basic idea of this approach is to reformulate the governing equation in a moving frame, and to derive a WKB-type function merely defined on the Lagrangian manifold induced by the Hamiltonian flow. From this WKB-type function, the wave solution can be retrieved to within first order accuracy by a coherent state integral. The merit of the proposed approach is manyfold. Firstly, compared with the thawed Gaussian beam approaches, it presents an approximate wave solution with first order asymptotic accuracy pointwise, even around caustics. Secondly, compared with the canonical operator method, this approach does not require any a priori knowledge about the structure of the Lagrangian manifold. Thirdly, compared with the frozen Gaussian beam approaches such as the Herman-Kluk semi-classical propagator method, the proposed approach involves an integral on a manifold of much lower dimension. We report numerical tests on both Schrödinger and Helmholtz equations.

Keywords

Hamiltonian system, coherent state, unitary representation, Lagrangian manifold, caustics, semi-classical approximation

2010 Mathematics Subject Classification

35F10, 65M25, 78M35

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Published 7 September 2012