Energy Dissipation Analysis of Implicit-Explicit Linear Multistep Methods for Gradient Flows Using a Simple Multiplier

Abstract. This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the original energy and a nonnegative quadratic modification. We first test IMEX-LMMs with a simple multiplier, the first-order time difference of numerical solutions. Then, it is shown that the associated nonnegative quadratic modification can be constructed if and only if two generating polynomials (corresponding to the LMM) are positive on [Formula: see text]. Based on this, the modified energy is proved to decay over time under a mild time-step restriction depending on the lower bounds of the associated generating polynomials. As a consequence, the energy dissipation of the well-known backward differentiation formula methods up to fifth order can be obtained straightforwardly. Furthermore, we construct for the first time (to the best of our knowledge) a sixth-order energy-dissipative IMEX-LMM and also prove the sixth-order barrier of energy-dissipative IMEX-LMMs when testing the simple multiplier. Some numerical experiments are conducted to verify our theoretical results.