Invariant domain preserving central schemes for nonlinear hyperbolic systems

We propose a central scheme framework for the approximation of hyperbolic systems of conservation laws in any space dimension. The new central schemes are defined so that any convex invariant set containing the initial data can be an invariant domain for the numerical method. The underlying first-order central scheme is the analog of the guaranteed maximum speed method of [J.‑L. Guermond and B. Popov, SIAM J. Anal., 54(4):2466–2489, 2016] adjusted to the finite volume framework. There are three novelties in this work. The first one is that any classical second-order central scheme can be modified to satisfy an invariant domain property of the first-order scheme via a process which we call convex limiting. This is done by using convex flux limiting along the lines of [J. ‑L. Guermond, B. Popov and I. Tomas, Comput. Meth. Appl. Mech. Engrg., 347:143–175, 2019]. The second novelty is the design of a new second-order method based on slope limiting only. The new local slope reconstruction technique is based on convex limiting so that the cell interface values are corrected to fit into a local invariant domain of the hyperbolic system. This new type of slope limiting depends on the hyperbolic system and to the best of our knowledge is the only one to guarantee local invariant domain preservation. Both schemes, flux and slope limiting based, are shown to be secondorder accurate for smooth solutions in the $L^\infty$-norm and robust in all test cases. The third novelty is a new second-order method based on the MAPR limiter from [I. Christov and B. Popov, J. Comput. Phys., 227(11):5736–5757, 2008] and adaptive slope limiting in the spirit of [A. Kurganov, G. Petrova and B. Popov, SIAM J. Sci. Comput., 29(6):2381–2401, 2007] but based on an entropy commutator. This new method can be used as an underlying high-order method and combined with convex flux limiting to guarantee a local invariant domain property. The time stepping of all methods is done by using strong stability preserving Runge–Kutta methods and the invariant domain property is proved under a standard CFL condition.

Keywords

nonlinear hyperbolic systems, Riemann problem, invariant domain, second-order method, convex limiting, finite volume method, central schemes

2010 Mathematics Subject Classification

35L65, 65M08, 65M12, 65M15

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This material is based upon work supported in part by the National Science Foundation grants DMS 1619892; by the Air Force Office of Scientific Research, USAF, under grant/contract number FA99550-12-0358; and by the Army Research Office under grant/contract number W911NF-15-1-0517.

Received 4 August 2019

Accepted 21 September 2020

Published 12 April 2021