IMEX methods for thin-film equations and Cahn-Hilliard equations with variable mobility

TL;DR

This work develops efficient, energy-stable numerical schemes for nonlinear fourth-order diffusion equations with variable mobility, including the Cahn-Hilliard equation and thin-film models. It combines a biharmonic-modified splitting (BHM) with implicit-explicit (IMEX) time stepping, using a linear constant-coefficient implicit term to enable linear solves and Fourier pseudo-spectral spatial discretization. The authors introduce and analyze BHM-IMEX1, BHM-IMEX2, and iterative BHM-BE_K, BHM-CN_K schemes, along with dynamic splitting where the fourth-order parameter $M_2$ adapts in time, and they perform extensive accuracy and energy-stability tests. The results show second-order accuracy for IMEX schemes, improved accuracy over the original BHM approach, and energy-decreasing behavior that is not guaranteed to be unconditional for variable mobility, highlighting the practical balance between stability and accuracy and providing guidance for parameter selection in simulations. These methods offer a scalable and implementable framework for long-time simulations of CHVM and TF problems, with potential extensions to higher dimensions and GPU architectures.

Abstract

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.

Figures (9)

• Figure 1: Thin film test problem 1: (left) The final state of the reference solution $U_*({\bf x},T)$ at $T=1$. (right) Convergence plots for the four non-iterative schemes (\ref{['BHMBE']}, \ref{['BHMCN']}, \ref{['imex']}, \ref{['imex2']}), with splitting parameters $M_2=0.32275, M_1=0$.
• Figure 2: Test problem 1 (continued): (left) Comparison of the IMEX schemes with the iterative BHM-BE$_J$ scheme for $J=1,2,4,8$, all with $M_2=0.32275$ and $M_1=0$. (right) Error curves for the BHM-BE$_J$ iterative methods with splitting parameters $M_2=0.07, M_1=0$.
• Figure 3: Test problem 1 (continued): The error for the time-stepping methods at fixed $h=0.125$ over a range of values for the $M_2$ splitting parameter. The curves terminate at minimum values $M_2^*$ below which simulations went unstable. Vertical lines at fixed values for the $\overline{\alpha}$ ratio in \ref{['aMmax']} are shown for reference.
• Figure 4: Test problem 2: (left) Error curves for the BHM-CN$_J$ iterative methods with $J=1,2,4,8$ and the two IMEX schemes, (right) Error curves for IMEX2 using fixed splitting parameter $M_2=\overline{\mathcal{M}}_{\max}$ compared with dynamic splitting with $M_{2,n}=\overline{\mathcal{M}}(t_n)$ and $M_{2,n}=0.9\overline{\mathcal{M}}(t_n)$.
• Figure 5: Test problem 3: Energy stability of numerical solutions of the Cahn-Hilliard equation \ref{['CHclassic']} using the BHM-BE$_1$ method. Simulations with $(h, M_1)$ parameters that are energy-stable are marked yellow, unstable parameters are shown in blue.