Hyperbolic model for Helmholtz equation with impedance boundary conditions

TL;DR

This paper introduces a hyperbolic reformulation of the Helmholtz equation with impedance boundary conditions, enabling finite-time steady states and efficient high-accuracy solutions for large wavenumbers using a well-balanced scheme.

Contribution

The authors propose a novel hyperbolic model for the Helmholtz equation with impedance boundary conditions, achieving finite-time steady states and high accuracy with fewer computational resources.

Findings

Finite-time convergence to steady state.

High accuracy for large wavenumbers.

Effective use of well-balanced scheme.

Abstract

Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations. Particularly interesting property of the proposed hyperbolic model is that steady state is achieved in finite time. For large wavenumber the numerically challenging task for Helmholtz equation is achieving high accuracy with small number of nodal points. We successfully solved this problem by means of using well balanced scheme approach. Numerical tests demonstrate excellent computational potential of the proposed method: high accuracy is achieved for large wavenumber with small number of nodal points in space and time.