High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices
type: Item
instance of: scholarly article
title: High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices
zbMATH Open document ID: 1398.65242
published in: Journal of Scientific Computing
publication date: 2018-01-16T00:00:00Z
review text: The author considers linear hyperbolic systems in \(d\geq 1\) space dimensions along with initial conditions and, on inflow boundaries, first order data in a form remembering the Roe averages. The aim is to avoid cumbersome inversion of global mass matrices when working with globally continuous finite element approximations in space. For this aim, a procedure of time stepping is presented which approximates the Picard iteration using a first-order operator (element-wise Euler forward) and gaining higher order with the help of a finite number of steps of deferred correction iteration defined using a higher-order explicit time integration method. The author shows the Lipschitz continuity of the spatial residuals appearing in the first-order operator and proves also the invertibility of the latter. He further clarifies conditions on the choice of basis functions for the spatial approximation (which exclude Lagrange elements). In his numerical experiments, results for one- and two-dimensional examples are reported which include also discontinuous solutions and a nonlinear case. Even though the paper is said to constitute an extended version of two earlier papers, the presentation contains rather superficial details both in the use of language (e.g., ``the last one for the one dimensional for the one dimensional case'', p. 479) and mathematically (e.g., in formula (9) on p. 465, the last component of the operator \(\mathcal L^1\) is defined as \(y_1-y_0-\Delta t \int_{t_{n,0}}^{t_{n,1}}{\mathcal I}_0(f;s)ds\) which means \(y_1-y_0-\xi_1\Delta t^2 f(y_0,t_{n,0})\) and also contradicts formula (8) which instead of \(\xi_1\Delta t^2 f(y_0,t_{n,0})\) exhibits \(\xi_0\Delta t f(y_0,t_{n,0})\) -- but both versions are wrong).
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 6826581
zbMATH Keywords: first-order linear hyperbolic systems
zbMATH Keywords: time step method
zbMATH Keywords: continuous finite elements
zbMATH Keywords: deferred correction
author: R. Abgrall
describes a project that uses: Gmsh
MaRDI profile type: Publication
full work available at URL: s10915-017-0498-4
OpenAlex ID: W2739470739
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DOI: 10.1007/S10915-017-0498-4
description: scientific article; zbMATH DE number 6826581