整体几何光学方法与高频波的数值模拟

Global geometrical optics method is a new semi-classical approximation to the solutions of linear wave equations in the high-frequency regime. This method was initially proposed by the applicant in 2012 for the scalar equations. It is well known that the classical geometrical optics method suffers from the existence of caustics, at which the rays intersect and the amplitude blows up. Global geometrical optics method completely overcomes the difficulty of caustics, and it presents an approximation of wavefield with uniformly asymptotic accuracy. This project aims at extending the scope of global geometrical optics method, strengthening its theoretical foundation and promoting its applications in numerical high-frequency waves. There are three concrete research issues: (1) extending the global geometrical optics method to vector-valued high-frequency wave equations;(2)setting up the semi-classical approximation of the wavefield on lower-dimensional manifold of the real space;(3)developing the numerical quadrature for highly oscillatory integrals on Lagrangian manifolds. As applications, we apply the theory of global geometrical optics method to various equations from mathematical physics, such as second-order linear wave equation, Schr?dinger equation, Helmholtz equation, Dirac equation, Maxwell equations and linear elastic wave equations.

整体几何光学方法是2012年申请人提出的一个新的求解高频线性标量波动方程的半经典近似方法。该方法完全克服了经典几何光学方法面临的焦散点困难，提供的波场近似具有空间整体一致的渐近精度。本项目旨在拓展整体几何光学方法的适用范围，加强这一方法的理论基础，推动这一方法在高频波数值模拟中的应用。研究的具体内容有三：发展向量值线性波动方程的整体几何光学方法；建立实空间低维流形上的半经典近似；设计拉格朗日流形上振荡函数积分的高效数值积分方法。项目组将把整体几何光学方法应用到二阶波动方程，Schr?dinger方程，Helmholtz方程，Dirac方程，Maxwell方程和线弹性波方程等数理方程的数值模拟。

高频波动方程的数值求解是个非常困难的问题。由于解的特征波长很小，直接数值求解需要大量的自由度去做函数逼近。色散误差的存在给这类方程的数值求解带来了更大的困难。发展基于渐进理论的渐进数值方法在很多情况尤其是考虑初值问题时是个更好的选择。传统的渐进方法称为几何光学方法，或者WKB方法。但这个方法面临着所谓焦散点存在的困难，很多情况下不能提供整体有效的波场近似。本项目系统地发展了扩展WKB方法这一新的高频波渐进近似理论。该理论适用于一般的线性高频波动方程，能提供具有整体一致精度的波场近似。