Journal ArticleDOI
A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions
TLDR
The 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, introducing a universal framework for treating boundary conditions of any type.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 th...read more
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Journal ArticleDOI
A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound
TL;DR: This work considers fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound, and demonstrates that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable.
Journal ArticleDOI
Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials
TL;DR: This work solves the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time using an implicit finite difference scheme and solving an elliptic equation at each time step by the method of difference potentials (MDP).
Journal ArticleDOI
High-order difference potentials methods for 1D elliptic type models
TL;DR: One-dimensional elliptic type models are employed as the starting point to develop and numerically test high-order accurate Difference Potentials Method (DPM) for variable coefficient elliptic problems in heterogeneous media.
Journal ArticleDOI
High order numerical simulation of the transmission and scattering of waves using the method of difference potentials
TL;DR: The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems and solves the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions.
Journal ArticleDOI
High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces
TL;DR: This work considers 2D elliptic models with material interfaces and develops efficient high-order accurate methods based on Difference Potentials for such problems as partial differential equations with varying coefficients and domains with irregular geometry.