Bound-preserving convex limiting for high-order 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.