Bound-preserving Flux Limiting for High-Order Explicit Runge-Kutta Time Discretizations of Hyperbolic Conservation Laws

Dmitri Kuzmin, Manuel Quezada de Luna, David I. Ketcheson, Johanna Grull

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We introduce a general framework for enforcing local or global maximum principles in high-order space-time discretizations of a scalar hyperbolic conservation law. We begin with sufficient conditions for a space discretization to be bound preserving (BP) and satisfy a semi-discrete maximum principle. Next, we propose a global monolithic convex (GMC) flux limiter which has the structure of a flux-corrected transport (FCT) algorithm but is applicable to spatial semi-discretizations and ensures the BP property of the fully discrete scheme for strong stability preserving (SSP) Runge–Kutta time discretizations. To circumvent the order barrier for SSP time integrators, we constrain the intermediate stages and/or the final stage of a general high-order RK method using GMC-type limiters. In this work, our theoretical and numerical studies are restricted to explicit schemes which are provably BP for sufficiently small time steps. The new GMC limiting framework offers the possibility of relaxing the bounds of inequality constraints to achieve higher accuracy at the cost of more stringent time step restrictions. The ability of the presented limiters to recognize undershoots/overshoots, as well as smooth solutions, is verified numerically for three representative RK methods combined with weighted essentially nonoscillatory (WENO) finite volume space discretizations of linear and nonlinear test problems in 1D. In this context, we enforce global bounds and prove preservation of accuracy for the linear advection equation.
Original languageEnglish (US)
JournalJournal of Scientific Computing
Volume91
Issue number1
DOIs
StatePublished - Mar 4 2022

Bibliographical note

KAUST Repository Item: Exported on 2022-04-27
Acknowledgements: The work of Dmitri Kuzmin and Johanna Grüll was supported by the German Research Association (DFG) under Grant KU 1530/23-1. The work of Manuel Quezada de Luna and David I. Ketcheson was funded by King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia.

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Theoretical Computer Science
  • Software
  • General Engineering

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