INVARIANT-DOMAIN-PRESERVING HIGH-ORDER TIME STEPPING: I. EXPLICIT RUNGE-KUTTA SCHEMES

We introduce a technique that makes every explicit Runge-Kutta (ERK) time stepping method invariant domain preserving and mass conservative when applied to high-order discretizations of the Cauchy problem associated with systems of nonlinear conservation equations. The key idea is that at each stage of the ERK scheme one computes a low-order update, a high-order update, both defined from the same intermediate stage, and then one applies the nonlinear, mass conservative limiting operator. The main advantage over the strong stability preserving (SSP) paradigm is more flexibility in the choice of the ERK scheme, thus allowing for less stringent restrictions on the time step. The technique is agnostic to the space discretization. It can be combined with continuous finite elements, discontinuous finite elements, and finite volume discretizations in space. Numerical experiments are presented to illustrate the theory.