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.

High-order difference potentials methods for 1D elliptic type models

“…We are concerned here with a 1D elliptic type interface problem of the form: 2) subject to the Dirichlet boundary conditions specified at the points x = 0 and x = 1:…”

“…Like the method in [15], and IIM, GFM and MIB, methods based on Di↵erence Potentials preserve the underlying accuracy of the schemes being used for the space discretization of the continuous PDEs in each domain/subdomain. But compared to [15], and to IIM and GFM, methods based on Di↵erence Potentials are not restricted by the type of the boundary or interface conditions (as long as the continuous problems are well-posed), see [22] or some example of the recent works [2], [23,27,4], ect. Furthermore, DPM is computationally e cient since any change of the boundary/interface conditions a↵ects only a particular component of the overall algorithm, and does not a↵ect most of the numerical algorithm (this property of the numerical method is crucial for computational and mathematical modeling of many applied problems).…”

“…Finally, Di↵erence Potentials approach is well-suited for the development of parallel algorithms, see [23,27,4] -examples of the second-order in space schemes based on Di↵erence Potentials idea for 2D interface/composite domain problems. The reader can consult [22,24] and [19,20] for a detailed theoretical study of the methods based on Di↵erence Potentials, and ( [22,24,14,25,28,26,16,2,23,27,3,4], etc) for the recent developments and applications of DPM.…”

High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces

“…Moreover, numerical schemes based on Difference Potentials are well-suited for the development of parallel algorithms (see [30,36,8], etc.). The reader can consult [32,33] and [28,29] for a detailed theoretical study of the methods based on Difference Potentials, and [32,33,11,20,34,39,35,22,5,30,36,7,8,10,1,2], etc. for the recent developments and applications of DPM.…”

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

“…Starting from the noticeable work [1], a family of high-order compact schemes are derived for variety of Helmholtz equations based on the differencing of governing equation to eliminate higher order derivatives in the discretization error, such as, the compact finite difference scheme for one dimension problems [2], the compact fourth order accuracy finite difference approximation for two and three dimension problems on the Cartesian coordinate [1,3] and polar coordinates [4,5]. Other papers have extended the HOC scheme to Helmholtz equation with non-homogeneous materials [6], with nonstandard boundary conditions [7] and with singular solutions [8]. Recently, more accurate compact sixth order methods are considered for solving the Helmholtz equation with a constant wave number [9,10,11] and a variable wave number [12].…”