We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials

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

High order numerical simulation of the transmission and scattering of waves using the method of difference potentials

https://www.math.utah.edu/~albright/AEMX_2015.pdf

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

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...

D. S. Britt

Papers

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