A Parameter-Dependent Hybrid Second-Order Implicit–explicit (Imex) Method for the Numerical Solution of Pdes

Implicit–Explicit (IMEX) time integration methods have become an important class of numerical schemes for solving time-dependent partial differential equations (PDEs) that exhibit both stiff and non-stiff dynamics. In this study, a parameter-dependent hybrid IMEX scheme is developed for the efficient numerical solution of stiff and non-stiff PDE systems. The proposed method combines an explicit two-step Adams–Bashforth approximation for the non-stiff terms with a parameter-weighted implicit formulation for the stiff components. The introduction of a free parameter provides additional flexibility for controlling stability and accuracy of the method. The resulting scheme forms a family of second-order IMEX time-integration methods that generalizes classical Crank–Nicolson–Adams–Bashforth approaches. Theoretical analysis of the method is carried out, including consistency, stability, and convergence properties. The stability function and absolute stability region of the proposed scheme are derived and analyzed. Numerical experiments involving both stiff and non-stiff PDE test problems, including diffusion and reaction–diffusion equations, are performed to demonstrate the effectiveness and accuracy of the method. The results confirm the second-order convergence of the scheme and illustrate that the parameter-dependent formulation provides improved stability characteristics compared to classical IMEX approaches. The proposed method therefore offers an efficient and flexible framework for the numerical solution of time-dependent PDE models arising in scientific and engineering applications