A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes

12/02/2020
by   Alexander Zlotnik, et al.
0

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove that then the solution norm grows exponentially in time making the scheme strongly non-dissipative and therefore impractical. Numerical results confirm this conclusion. In addition, for some sequences of refining spatial meshes, an excessively strong condition between steps in time and space is necessary (even for the non-uniform in time stability) which is familiar for explicit schemes in the parabolic case.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

11/28/2020

On compact 4th order finite-difference schemes for the wave equation

We consider compact finite-difference schemes of the 4th approximation o...
01/26/2021

On Properties of Compact 4th order Finite-Difference Schemes for the Variable Coefficient Wave Equation

We consider an initial-boundary value problem for the n-dimensional wave...
05/15/2021

On properties of an explicit in time fourth-order vector compact scheme for the multidimensional wave equation

An initial-boundary value problem for the n-dimensional wave equation is...
05/06/2021

A mollifier approach to regularize a Cauchy problem for the inhomogeneous Helmholtz equation

The Cauchy problem for the inhomogeneous Helmholtz equation with non-uni...
02/08/2021

Understand Slope Limiter – Graphically

In this article, we illustrate how the concept of slope limiter can be i...
10/12/2019

On the numerical approximations of the periodic Schrödinger equation

We consider semidiscrete finite differences schemes for the periodic Scr...
09/26/2020

Uniform convergence and stability of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers' equation

In the paper, a newly developed three-point fourth-order compact operato...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.