A Compact Two-Stage Fourth-Order Two-Derivative IMEX Method with Mixed Compatibility, High Implicit-Solve Efficiency,and Enhanced Stiff Decay

Abstract

For additively split stiff evolution problems, classical fourth-order IMEX Runge-Kutta methods usually require several stages and complicated coupling order conditions. This paper proposes a compact fourth-order IMEX-type method based on a two-derivative formulation and analyzes its accuracy and stability properties. The proposed scheme incorporates the mixed explicit-implicit interaction directly through temporal derivatives evaluated along the full vector field, which ensures mixed compatibility for non-commuting split systems. With only one intermediate stage and two implicit solves per time step, the method achieves fourth-order accuracy while improving the accuracy obtained per implicit solve compared with classical multi-stage fourth-order IMEX-RK schemes. In addition, the method exhibits stronger damping of stiff modes in the purely implicit scalar limit and in the strong implicit-stiffness limit with a fixed explicit component. In the purely implicit scalar limit, its stability factor decays quadratically as the stiffness increases, whereas a representative classical fourth-order IMEX-RK reference method shows only linear decay. Numerical experiments on non-commuting split systems, scalar stiff-mode damping, increasing stiffness tests, and one- and two-dimensional advection-diffusion high-mode problems confirm the mixed consistency, the predicted stiff decay, and the smaller errors obtained under equal implicit-solve budgets in strongly stiff regimes.