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Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  31 October 2017

Elliott S. Wise ,
Ben T. Cox  and
Bradley E. Treeby
Elliott S. Wise*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
Ben T. Cox*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
Bradley E. Treeby*
Affiliation:
Department of Medical Physics and Biomedical Engineering, University College London, 2–10 Stephenson Way, London, NW1 2HE, United Kingdom
*
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
*Corresponding author. Email addresses:elliott.wise.14@ucl.ac.uk(E. S. Wise), b.cox@ucl.ac.uk(B. T. Cox), b.treeby@ucl.ac.uk(B. E. Treeby)
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Abstract

Moving mesh methods provide an efficient way of solving partial differential equations for which large, localised variations in the solution necessitate locally dense spatial meshes. In one-dimension, meshes are typically specified using the arclength mesh density function. This choice is well-justified for piecewise polynomial interpolants, but it is only justified for spectral methods when model solutions include localised steep gradients. In this paper, one-dimensional mesh density functions are presented which are based on a spatially localised measure of the bandwidth of the approximated model solution. In considering bandwidth, these mesh density functions are well-justified for spectral methods, but are not strictly tied to the error properties of any particular spatial interpolant, and are hence widely applicable. The bandwidth mesh density functions are illustrated in two ways. First, by applying them to Chebyshev polynomial approximation of two test functions, and second, through use in periodic spectral and finite-difference moving mesh methods applied to a number of model problems in acoustics. These problems include a heterogeneous advection equation, the viscous Burgers’ equation, and the Korteweg-de Vries equation. Simulation results demonstrate solution convergence rates that are up to an order of magnitude faster using the bandwidth mesh density functions than uniform meshes, and around three times faster than those using the arclength mesh density function.

Keywords

Moving mesh methodpseudospectral methodbandwidth

MSC classification

Secondary: 65M06: Finite difference methods 65M50: Mesh generation and refinement 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1286 - 1308
Copyright
Copyright © Global-Science Press 2017 

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