Spectrum-free and meshless solvers of parabolic PDEs
type: Item
instance of: scholarly article
title: Spectrum-free and meshless solvers of parabolic PDEs
zbMATH Open document ID: 1503.65267
published in: Journal of Scientific Computing
publication date: 2021-08-31T00:00:00Z
review text: The paper presents a collocation method for solving parabolic partial differential equations, e.g., the inhomogeneous heat equation, in two and three space dimensions. In space, the solution is discretized into a series of radial basis functions, e.g., Gaussian kernels or multi-quadratic kernels. For time integration,the semi-group of the heat equation is used and the corresponding exponential integrator is discretized with a Padé-Chebychev method. Moreover, for comparison, a linear combination of the implicit and explicit Euler method is used, i.e., a so-called \(\theta\) method, which corresponds to the Crank-Nicolson method for \(\theta=1/2\). Convergence of the time integration using the Padé-Chebychev method is provided and the existence of the spatial derivatives of the radial basis functions is analyzed. Then numerical properties and test computations are carried out and the computational cost is assessed. Convection-diffusion problems are likewise studied. Numerical results in 2D and 3D are reported for regular and irregular sampling in space and the errors with respect to a reference solution are determined. It is shown that the Padé-Chebychev method yields higher stability and better accuracy compared to previous results in the literature.
reviewed by: Kai Schneider
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65D12
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35Q79
zbMATH DE Number: 7388734
zbMATH Keywords: parabolic PDEs
zbMATH Keywords: heat equation
zbMATH Keywords: collocation methods
zbMATH Keywords: radial basis functions
zbMATH Keywords: spectrum
zbMATH Keywords: spectrum-free solvers
author: Giuseppe Patanè
describes a project that uses: Algorithm 368
describes a project that uses: Expokit
MaRDI profile type: Publication
full work available at URL: s10915-021-01604-x
OpenAlex ID: W3194759768
Wikidata QID: Q114225593
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DOI: 10.1007/S10915-021-01604-X
description: scientific article; zbMATH DE number 7388734