  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.
