Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion

Abstract

Finite difference techniques are widely used for the numerical simulation of time-dependent partial differential equations. In order to get better accuracy at low computational cost, researchers have attempted to develop higher order methods by improving other lower order methods. However, these types of methods usually suffer from a high degree of numerical dispersion. In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Padé based (CPD) and non-compact Padé based (NCPD) schemes for the acoustic wave equation. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. The effects of Courant Friedrichs Lewy (CFL) number, propagation angle and number of cells per wavelength on dispersion are studied.

Benchmarking Solvers for the One Dimensional Cubic Nonlinear Klein Gordon Equation on a Single Core

“…for which time stepping error comes from approximating u tt . As explained in among other places, [1], it is possible to approximate the leading order error term,…”

“…The solution of linear systems of equations is a time consuming process in numerical simulation of partial differential equations, and has motivated a number of benchmarks [2,4,6,11]. The solution of many partial differential equations requires a choice of discretization methods, in space and typically also in time, each of which presents numerous choices, each of which may have different relative performance on different computer architectures [1,5,9,10,15,[18][19][20]24]. The Klein Gordon equation is chosen as a mini-application because it is relatively simple, can be used to evaluate different time stepping methods and spatial discretization methods, and is representative of seismic wave solvers, and weather codes, all of which use a large amount of high performance computing time [1,10,21,28].…”

“…The solution of many partial differential equations requires a choice of discretization methods, in space and typically also in time, each of which presents numerous choices, each of which may have different relative performance on different computer architectures [1,5,9,10,15,[18][19][20]24]. The Klein Gordon equation is chosen as a mini-application because it is relatively simple, can be used to evaluate different time stepping methods and spatial discretization methods, and is representative of seismic wave solvers, and weather codes, all of which use a large amount of high performance computing time [1,10,21,28]. As a prelude to a three dimensional study of parallel solvers, a comparison of solvers for the one dimensional Klein Gordon equation on two architectures is presented showing the effects of discretization method on time to solution for a specified accuracy on a single core.…”

Alternating direction implicit time integrations for finite difference acoustic wave propagation: Parallelization and convergence

“…Above, the dependent variables are the scalar pressure u and the particle velocity vector v = (v, w) , while f may correspond to an additional forcing term. Appropriate initial and boundary conditions must be considered in order to find a particular solution to (1), and this velocity-pressure formulation is well suited to state boundary conditions of relevance in seismic applications. For instance, a free surface can be modeled by a homogeneous Dirichlet condition on u, and most absorbing conditions require the computation of u along the domain boundaries.…”

“…However, pressure is known along boundary edges because of Dirichlet conditions, thus this formulation also makes use of the reduced matrix Ū that only holds interior grid values. Time updating of these inner discrete pressures is based on the discretization of momentum conservation in (1), and requires approximations to v x and w y . In space, this discretization in carried out by reduced operators P and Q, also introduced in Appendix A, applied to discrete velocities Vx PT = V QT ,…”