Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws


Authors: Sulin Wang and Zhengfu Xu
Journal: Math. Comp. 88 (2019), 691-716
MSC (2010): Primary 58J45, 65M06
DOI: https://doi.org/10.1090/mcom/3364
Published electronically: June 13, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we focus on developing locally conservative high order finite difference methods with provable total variation stability for solving one-dimensional scalar conservation laws. We introduce a new criterion for designing high order finite difference schemes with provable total variation stability by measuring the total variation of an expanded vector. This expanded vector is created from grid values at $ t^{n+1}$ and $ t^n$ with ordering determined by upwinding information. Achievable local bounds for grid values at $ t^{n+1}$ are obtained to provide a sufficient condition for the total variation of the expanded vector not to be greater than total variation of the initial data. We apply the Flux-Corrected Transport type of bound preserving flux limiters to ensure that numerical values at $ t^{n+1}$ are within these local bounds. When compared with traditional total variation bounded high order methods, the new method does not depend on mesh-related parameters. Numerical results are produced to demonstrate: the total variation of the numerical solution is always bounded; the order of accuracy is not sacrificed. When the total variation bounded flux limiting method is applied to a third order finite difference scheme, we show that the third order of accuracy is maintained from the local truncation error point of view.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 58J45, 65M06

Retrieve articles in all journals with MSC (2010): 58J45, 65M06


Additional Information

Sulin Wang
Affiliation: Department of Mathematical Science, Michigan Tech University, Houghton, Michigan 49931
Email: sulinw@mtu.edu

Zhengfu Xu
Affiliation: Department of Mathematical Science, Michigan Tech University, Houghton, Michigan 49931
Email: zhengfux@mtu.edu

DOI: https://doi.org/10.1090/mcom/3364
Keywords: Hyperbolic conservation laws, bound preserving, flux limiters, high order scheme, total variation stability
Received by editor(s): March 11, 2017
Received by editor(s) in revised form: May 17, 2017, and November 13, 2017
Published electronically: June 13, 2018
Additional Notes: The authors would like to acknowledge the support of the NSF grant DMS-1316662 “High Order Maximum Principle Preserving Finite Difference Schemes for Hyperbolic Conservation Laws”.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society