Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods

Abstract

We propose entropy-preserving and entropy-stable partitioned Runge-Kutta (RK) methods. In particular we develop entropy conditions for implicit-explicit methods and a class of second-order multirate methods. We extend relaxation ideas for explicit methods to partitioned RK methods. We show that the proposed methods support fully entropy-preserving and entropy-stability properties at a discrete level. Numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the behavior of these methods.

Electron dynamics in extended systems within real-time time-dependent density functional theory

“…These methods are referred to as relaxation RK and rely on modifying the prescribed time step (typically reducing it by a small fraction) so that the solution at each of these modified steps preserves energy. 70,71 Explicit methods are conditionally stable; nevertheless, the stability regions can be optimized for a specific eigenvalue portrait, which is a promising strategy to improve their performance. Furthermore, new machine learning developments in neural ODE may provide new ways to accelerate the time stepping process.…”