Asymptotic expansions of the Helmholtz equation solutions using approximations of the Dirichlet to Neumann operator

TL;DR

This paper develops new asymptotic expansions for the Helmholtz equation solutions by approximating the Dirichlet to Neumann operator, improving high-frequency numerical solver accuracy in shadow regions.

Contribution

It introduces first and second order approximations of the Dirichlet to Neumann operator to derive enhanced asymptotic expressions for Helmholtz solutions.

Findings

New asymptotic expansions for the normal derivative of the Helmholtz solution.

Improved accuracy of high-frequency numerical solvers in shadow regions.

Enhanced ansatz design for integral equation-based methods.

Abstract

This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This work uses first and second order approximations of this operator to derive new asymptotic expressions of the normal derivative of the total field. The resulting expansions can be used to appropriately choose the ansatz in the design of high-frequency numerical solvers, such as those based on integral equations, in order to produce more accurate approximation of the solutions around the shadow and deep shadow regions than the ones based on the usual ansatz.