 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, 73(2-3):461-494.

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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