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High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices


Abgrall, Rémi (2017). High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. Journal of Scientific Computing:Epub ahead of print.

Abstract

When integrating unsteady problems using globally continuous representation of the solution, as for continuous finite element methods, one faces the problem of inverting a mass matrix. In some cases, one has to recompute this mass matrix at each time steps. In some other methods that are not directly formulated by standard variational principles, it is not clear how to write an invertible mass matrix. Hence, in this paper, we show how to avoid this problem for hyperbolic systems, and we also detail the conditions under which this is possible. Analysis and simulation support our conclusions, namely that it is possible to avoid inverting mass matrices without sacrificing the accuracy of the scheme. This paper is an extension of Abgrall et al. (in: Karasözen B, Manguoglu M, Tezer-Sezgin M, Goktepe S, Ugur O (eds) Numerical mathematics and advanced applications ENUMATH 2015. Lecture notes in computational sciences and engineering, vol 112, Springer, Berlin, 2016) and Ricchiuto and Abgrall (J Comput Phys 229(16):5653–5691, 2010). © 2017 Springer Science+Business Media, LLC

Abstract

When integrating unsteady problems using globally continuous representation of the solution, as for continuous finite element methods, one faces the problem of inverting a mass matrix. In some cases, one has to recompute this mass matrix at each time steps. In some other methods that are not directly formulated by standard variational principles, it is not clear how to write an invertible mass matrix. Hence, in this paper, we show how to avoid this problem for hyperbolic systems, and we also detail the conditions under which this is possible. Analysis and simulation support our conclusions, namely that it is possible to avoid inverting mass matrices without sacrificing the accuracy of the scheme. This paper is an extension of Abgrall et al. (in: Karasözen B, Manguoglu M, Tezer-Sezgin M, Goktepe S, Ugur O (eds) Numerical mathematics and advanced applications ENUMATH 2015. Lecture notes in computational sciences and engineering, vol 112, Springer, Berlin, 2016) and Ricchiuto and Abgrall (J Comput Phys 229(16):5653–5691, 2010). © 2017 Springer Science+Business Media, LLC

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:18 July 2017
Deposited On:09 Oct 2017 16:38
Last Modified:10 Oct 2017 08:01
Publisher:Springer
ISSN:0885-7474
Publisher DOI:https://doi.org/10.1007/s10915-017-0498-4

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