Bound-preserving flux limiting for high-order explicit Runge-Kutta time discretizations of hyperbolic conservation laws

Abstract

We introduce a general framework for enforcing local or global inequality constraints in high-order time-stepping methods for a scalar hyperbolic conservation law. The proposed methodology blends an arbitrary Runge-Kutta scheme and a bound-preserving (BP) first-order approximation using two kinds of limiting techniques. The first one is a predictor-corrector method that belongs to the family of flux-corrected transport (FCT) algorithms. The second approach constrains the antidiffusive part of a high-order target scheme using a new globalized monolithic convex (GMC) limiter. The flux-corrected approximations are BP under the time step restriction of the forward Euler method in the explicit case and without any time step restrictions in the implicit case. The FCT and GMC limiters can be applied to antidiffusive fluxes of intermediate RK stages and/or of the final solution update. Stagewise limiting ensures the BP property of intermediate cell averages. If the calculation of high-order fluxes involves polynomial reconstructions from BP data, these reconstructions can be constrained using a slope limiter to correct unacceptable input. The BP property of the final solution is guaranteed for all flux-corrected methods. Numerical studies are performed for one-dimensional test problems discretized in space using explicit weighted essentially nonoscillatory (WENO) finite volume schemes.

Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

“…The final stage of a general high-order Runge-Kutta method for (3.3) can also be written in the form (3.5) and constrained to produce ū * i ∈ G; see [33] for details.…”

“…The high-order Runge-Kutta time discretization of (4.1) is recovered in the case α n+1 ij = 1 ∀j ∈ N i \{i}. In the process of flux correction for the final stage (4.17), the IDP property is enforced (as in [33]) using the MCL limiter for f n+1 ij . To prevent a loss of accuracy at this stage, the bounds should be global (as in [33]) or defined using all states u n j that may influence u n+1 i if time integration is performed using the high-order Runge-Kutta scheme.…”

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

“…Many problems in nature are described either by ordinary differential equations (ODEs) or partial differential equations (PDEs) and the numerical methods that approximate their solutions should preserve the physical quantities of the underlying problem. To keep the positivity for example Patankar approaches [32,21,28] or adaptive/limiting strategies [25,29] can be found in the literature, while, recently, Ketcheson proposed relaxation Runge-Kutta (RRK) methods to guarantee conservation or stability with respect to any innerproduct norm. In a series of papers, he and collaborators have further extended the relaxation approach to multistep schemes and applied it to different kind of problems, cf.…”