Volume 22, Issue 5
Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods

Elliott S. Wise, Ben T. Cox & Bradley E. Treeby

Commun. Comput. Phys., 22 (2017), pp. 1286-1308.

Published online: 2017-11

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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.

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@Article{CiCP-22-1286, author = {}, title = {Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods}, journal = {Communications in Computational Physics}, year = {2017}, volume = {22}, number = {5}, pages = {1286--1308}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0246}, url = {http://global-sci.org/intro/article_detail/cicp/10443.html} }
TY - JOUR T1 - Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods JO - Communications in Computational Physics VL - 5 SP - 1286 EP - 1308 PY - 2017 DA - 2017/11 SN - 22 DO - http://doi.org/10.4208/cicp.OA-2016-0246 UR - https://global-sci.org/intro/article_detail/cicp/10443.html KW - AB -

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.

Elliott S. Wise, Ben T. Cox & Bradley E. Treeby. (2020). Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods. Communications in Computational Physics. 22 (5). 1286-1308. doi:10.4208/cicp.OA-2016-0246
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