A plane wave method based on approximate wave directions for two dimensional Helmholtz equations with large wave numbers

TL;DR

This paper introduces an efficient high-accuracy plane wave method for solving 2D Helmholtz equations with large wave numbers by using adaptive basis functions based on approximate wave directions, improving computational efficiency.

Contribution

The paper develops an adaptive plane wave space with small dimensions, determined by approximate wave directions, and provides theoretical and numerical analysis of its approximation capabilities.

Findings

Achieves high accuracy in wave direction computation

Provides a best $L^2$ approximation for solutions with large wave numbers

Numerical results demonstrate the method's efficiency

Abstract

In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an "adaptive" plane wave space with small dimensions, in which each plane wave basis function is determined by such an approximate wave direction. We establish a best $L^2$ approximation of the plane wave space for the analytic solutions of homogeneous Helmholtz equations with large wave numbers and report some numerical results to illustrate the efficiency of the proposed method.