Mesh Density Functions Based on Local Bandwidth Applied to Moving Mesh Methods (Communications in Computational Physics, 2017)
Moving mesh methods provide an efficient way of solving partial differentialequations for which large, localised variations in the solution necessitate locallydense spatial meshes. In one-dimension, meshes are typically specified using the arclengthmesh density function. This choice is well-justified for piecewise polynomialinterpolants, but it is only justified for spectral methods when model solutions includelocalised steep gradients. In this paper, one-dimensional mesh density functions arepresented which are based on a spatially localised measure of the bandwidth of the approximatedmodel solution. In considering bandwidth, these mesh density functionsare well-justified for spectral methods, but are not strictly tied to the error propertiesof any particular spatial interpolant, and are hence widely applicable. The bandwidthmesh density functions are illustrated in two ways. First, by applying them toChebyshev polynomial approximation of two test functions, and second, through usein periodic spectral and finite-difference moving mesh methods applied to a numberof model problems in acoustics. These problems include a heterogeneous advectionequation, the viscous Burgers’ equation, and the Korteweg-de Vries equation. Simulationresults demonstrate solution convergence rates that are up to an order of magnitudefaster using the bandwidth mesh density functions than uniform meshes, andaround three times faster than those using the arclength mesh density function.