INVARIANT-DOMAIN PRESERVING HIGH-ORDER TIME STEPPING: II. IMEX SCHEMES

We consider high-order discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part with diffusion and stiff relaxation terms. We propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariantdomain preserving and mass conservative. Following the ideas introduced in Part I on explicit Runge- Kutta schemes, the IMEX scheme is written in incremental form. At each stage, we first combine a low-order and a high-order hyperbolic update using a limiting operator, then we combine a low-order and a high-order parabolic update using another limiting operator. The proposed technique, which is agnostic to the space discretization, allows one to optimize the time step restrictions induced by the hyperbolic substep. To illustrate the proposed methodology, we derive four novel IMEX methods with optimal efficiency. All the implicit schemes are singly diagonal. One of them is A-stable and the other three are L-stable. The novel IMEX schemes are evaluated numerically on systems of stiff ordinary differential equations and nonlinear conservation equations.