Summation by parts operators for finite difference approximations of second derivatives
type: Item
instance of: scholarly article
title: Summation by parts operators for finite difference approximations of second derivatives
zbMATH Open document ID: 1071.65025
author: Ken Mattsson
author: Jan Nordström
published in: Journal of Computational Physics
publication date: 2004-10-12T00:00:00Z
review text: The authors design central difference operators to approximate the derivatives \(d/dx\), \(d^2/dx^2\) at evenly spaced points in an interval \([a,b]\) which satisfy numerical formulas analogous to the integration by parts formulas from calculus (summation by parts (SBP) formulas). Using discrete diagonal norms the authors construct third, fourth, and fifth degree accurate SBP operators approximating \(d/dx\), \(d^2/dx^2\) and using discrete matrix norms fourth, sixth, and eighth accurate approximations. These approximations lead to easily obtained energy estimates for mixed hyperbolic parabolic problems in \((x,t)\) variables provided that the initial-boundary conditions are imposed such that the SBP property is preserved. For this purpose the authors employ a method referred to as the SAT method for boundary conditions [\textit{M. Carpenter}, \textit{D. Gottlieb} and \textit{S. Abarbanel}, J. Comput. Phys. 111, No. 2, 220--236 (1994; Zbl 0832.65098)]. Energy estimates and an error analysis are made for the convection diffusion equation and a model for the compressible Navier-Stokes equation. To obtain \(2p\)-order of accuracy it is proved that the approximating second derivatives should have order \(2p\) in the interior region and \(2p-2\) near the boundary.
Mathematics Subject Classification ID: 65D25
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 2107334
zbMATH Keywords: High order finite difference methods
zbMATH Keywords: Numerical stability
zbMATH Keywords: Second derivatives
zbMATH Keywords: Accuracy
zbMATH Keywords: Boundary conditions
zbMATH Keywords: summation by parts formulas
zbMATH Keywords: mixed hyperbolic parabolic problems
zbMATH Keywords: error analysis
zbMATH Keywords: convection diffusion equation
zbMATH Keywords: compressible Navier-Stokes equation
reviewed by: John B. Butler jun.
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.jcp.2004.03.001
OpenAlex ID: W2051465806
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DOI: 10.1016/J.JCP.2004.03.001
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description: scientific article; zbMATH DE number 2107334