Average Energy Dissipation Rates of Implicit-Explicit Runge-Kutta Methods for Gradient Flow Problems

A unified theoretical framework is suggested to examine the energy dissipation properties at all stages of additive implicit-explicit Runge-Kutta (IERK) methods up to fourth-order accuracy for gradient flow problems. We construct some parameterized IERK methods by applying the so-called first same as last method, that is, the diagonally implicit Runge-Kutta method with the explicit first stage and stiffly-accurate assumption for the linear stiff term, and applying the explicit Runge-Kutta method for the nonlinear term. The main part of our framework is to construct the differential forms and the associated differentiation matrices of IERK methods by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an IERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. The recent indicator, namely average energy dissipation rate, is also adopted for these multi-stage methods to evaluate the overall energy dissipation rate of an IERK method such that one can choose proper parameters in some parameterized IERK methods. A key feature of IERK methods is that the differentiation matrices and average energy dissipation rates are linearly dependent on the time-space discretization parameters compared with those of explicit exponential Runge-Kutta methods in [Math. Comput., 94:1721–1759, 2025]. It is found that the selection of method parameters in the IERK methods is at least as important as the selection of different IERK methods. Extensive numerical experiments are also included to support our theory.