On coordinate transformations for summation-by-parts operators
type: Item
instance of: scholarly article
title: On coordinate transformations for summation-by-parts operators
zbMATH Open document ID: 1057.65054
DOI: 10.1023/A:1025881528802
published in: Journal of Scientific Computing
publication date: 2004-05-27T00:00:00Z
review text: Many problems in computational fluid dynamics require very accurate spatial resolution to correctly model high frequency phenomena. The author examines properties of a special class of high order finite difference operators obeying a summation-by-parts rule (SBP-operators) developed by \textit{H. O. Kreiss} and \textit{G. Scherer} in [Finite element and finite difference methods for hyperbolic partial differential equations. Math. Aspects Finite Elem. Partial Differ. Equ., Proc. Symp. Madison 1974, 195--212 (1974; Zbl 0355.65085) and On the existence of energy estimates for difference approximations for hyperbolic systems (1977)]. These norms are of a non-diagonal form. The main advantage of SBP-operators is that they simplify the procedure to obtain energy estimates. To accurately model the influence of realistic geometries, it is essential to be able to use the SBP-operators on curvilinear grids. In the paper accurate coordinate transformation operators are analyzed using the linear advection equation as a model problem. So, the objective of the paper is to study how coordinate transformations influence high order SBP schemes and if possible, construct an accurate SBP preserving coordinate transformation operator. But, as is stated in Theorem 3.2, the conditions for a sufficiently accurate SBP preserving coordinate operators are not possible to fulfill. Thus, either the stability or the accuracy conditions must be relaxed. If the accuracy conditions are kept (using a block norm) there is no systematic way of proving stability and the location of the eigenvalues has to be checked for each problem. Further, there is nothing that guarantees stability when the number of grid points are increased. If the accuracy conditions are relaxed, a diagonal norm scheme was shown to be the only choice that fulfill the remaining stability conditions making it possible to treat the boundaries in a stable and systematic manner. The contents of the paper report are divided as follows…
reviewed by: Leonid B. Chubarov
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 76M20
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 2069185
zbMATH Keywords: high order finite difference operators
zbMATH Keywords: equidistant grid
zbMATH Keywords: energy estimates
zbMATH Keywords: curvilinear grids
zbMATH Keywords: coordinate transformation operators
zbMATH Keywords: stability
zbMATH Keywords: accuracy
zbMATH Keywords: numerical experiments
zbMATH Keywords: summation-by-parts operators
zbMATH Keywords: linear advection equation
author: M. Svärd
MaRDI profile type: Publication
full work available at URL: a:1025881528802
OpenAlex ID: W1495503452
description: scientific article; zbMATH DE number 2069185