We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn-Hilliard models with variable mobility. This splitting method incorporates a linear, constant coefficient implicit step, facilitating efficient computational implementation. We investigate the influence of stabilizing splitting parameters on the numerical solution computationally, considering various initial conditions. Furthermore, we generate energy-stability plots for the proposed methods, examining different choices of splitting parameter values and timestep sizes. These methods enhance the accuracy of the original bi-harmonic-modified (BHM) approach, while preserving its energy-decreasing property and achieving second-order accuracy. We present numerical experiments to illustrate the performance of the proposed methods.

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