A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions

Abstract

We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate the resulting numerical capabilities by solving a range of nonstandard boundary value problems for the Helmholtz equation. These include problems with variable coefficient Robin boundary conditions (including discontinuous coefficients) and problems with mixed (Dirichlet/Neumann) boundary conditions. In all our simulations, we use a Cartesian grid and a circular boundary curve. For those test cases where the overall solution was smooth, our methodology has consistently demonstrated the design fourth-order rate of grid convergence, whereas when the regularity of the solution was not sufficient, the convergence slowed down, as expected. We also show that every additional boundary condition requires only an incremental additional expense.