Mesh density functions based on local bandwidth applied to moving mesh methods

摘要

Moving mesh methods provide an efficient way of solving partial differentialequations for which large, localised variations in the solution necessitatelocally dense spatial meshes. In one-dimension, meshes are typically specifiedusing the arclength mesh density function. This choice is well-justified forpiecewise polynomial interpolants, but it is only justified for spectralmethods when model solutions include localised steep gradients. In this paper,one-dimensional mesh density functions are presented which are based on aspatially localised measure of the bandwidth of the approximated modelsolution. In considering bandwidth, these mesh density functions arewell-justified for spectral methods, but are not strictly tied to the errorproperties of any particular spatial interpolant, and are hence widelyapplicable. The bandwidth mesh density functions are demonstrated by applyingperiodic spectral and finite-difference moving mesh methods to a number ofmodel problems in acoustics. These problems include a heterogeneous advectionequation, the viscous Burgers' equation, and the Korteweg-de Vries equation.Simulation results demonstrate solution convergence rates that are up to anorder of magnitude faster using the bandwidth mesh density functions thanuniform meshes, and around three times faster than those using the arclengthmesh density function.