Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws

TL;DR

This paper introduces a novel limiting technique that preserves invariant domains in high-order numerical schemes for nonlinear hyperbolic conservation laws, applicable to various discretizations and time integration methods.

Contribution

It generalizes flux-corrected transport limiters to systems of conservation laws, ensuring invariant domain preservation across explicit and implicit schemes.

Findings

The method effectively preserves invariant domains in scalar and Euler equations.

It can be integrated with finite volume and discontinuous Galerkin discretizations.

Numerical experiments demonstrate improved stability and accuracy.

Abstract

This work concerns the design and analysis of a limiting technique that allows the preservation of invariant domains for high-order numerical approximations of nonlinear hyperbolic systems of conservation laws. The method can be applied to any conservative discretization method in space as well as to a wide range of explicit and implicit time integration schemes. The method limits the high-order solution around a low-order accurate solution that is known to preserve all the invariant domains. It generalizes the flux-corrected transport limiter [J. P. Boris and D. L. Book, J. Comput. Phys., 11, 1973; S. T. Zalesak, J. Comput. Phys., 31, 1979] to systems of conservation laws and relies on the limitation of antidiffusive fluxes, but defines the limiting coefficients so as to express the limited solution as a convex combination of invariant domain preserving quantities similarly to the convex limiting framework [Guermond et al., Comput. Methods Appl. Mech. Engrg., 347, 2019]. We give details on the derivation of this limiting technique and provide some illustration with finite volume or discontinuous Galerkin (DG) space discretizations associated to explicit or implicit Runge-Kutta methods as well as to time DG integrations. The limiter is applied iteratively to refine the limited solution around the high-order one, while preserving the invariant domains, and a heuristic is proposed to accelerate its convergence. Numerical experiments solving one- and two-dimensional problems involving scalar hyperbolic equations and the compressible Euler equations are presented to illustrate the properties of these schemes.