Extended WKB Analysis For The High-Frequency Vectorial Linear Wave Equations

Solving high-frequency wave equation is a classical problem in physics and mathematics.For most high-frequency wave equations,it is hard to find the exact solution.Therefore,we need to develop efficient numerical methods.In principle,traditional numerical methods for PDEs could be used to solve this kind of problems.However,the highly oscillatory nature of wave fields forces us to use very small grids,which presents a great numerical challenge.To develop more efficient numerical methods,various asymptotic theories have been developed.Among them,the WKB method,also called the geometrical optics method,is a classical choice.It is well-known that the WKB method faces the difficulty due to the appearance of caustics.To solve this problem,Zheng developed the extended WKB method for high-frequency scalar wave equation.Different from the classical WKB method,the phase function and the amplitude function in the extended WKB method are defined on a displaced Lagrangian submanifold Λ,and the proposed asymptotic solution is expressed as an integral of a family of Gaussian functions which are parametrized by Λ.Since the extended WKB method does not depend on the coordinate of the Lagrangian submanifold in the phase space,this method is capable of solving the caustic problem.This dissertation develops the extended WKB method for the high-frequency vectorial wave equations.Assume u is an extended WKB function,and H(W)is the associated Weyl’s quantized operator for the high-frequency vectorial wave equation.We prove that H(W)u is still an extended WKB function under suitable restriction to the Lagrangian submanifold Λ.After performing an asymptotic analysis analogous to the WKB method,we deduce an ODE system for the phase and the amplitude functions.Choosing a proper ODE solver and a proper numerical quadrature to compute the involved integral,we then derive an asymptotic numerical solution for the highfrequency vectorial wave equation.In the last part of this dissertation,we validate the effectiveness of the proposed extended WKB method through some numerical tests.