2023/12/25 by Xinyu Cheng, Cheng, Xinyu
Materials Science · #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2312.15656
openalex publication_date 2023/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Phase field models are gradient flows with their energy naturally dissipating in time. In order to preserve this property, many numerical schemes have been well-studied. In this paper we consider a well-known method, namely the exponential integrator method (EI). In the literature a few works studied several EI schemes for various phase field models and proved the energy dissipation by either requiring a strong Lipschitz condition on the nonlinear source term or certain L^∞ bounds on the numerical solutions (maximum principle). However for phase field models such as the (non-local) Cahn-Hilliard equation, the maximum principle no longer exists. As a result, solving such models via EI schemes remains open for a long time. In this paper we aim to give a systematic approach on applying EI-type schemes to such models by solving the Cahn-Hilliard equation with a first order EI scheme and showing the energy dissipation. In fact second order EI schemes can be handled similarly and we leave the discussion in a subsequent paper. To our best knowledge, this is the first work to handle phase field models without assuming any strong Lipschitz condition or L^∞ boundedness. Furthermore, we will analyze the L2 error and present some numerical simulations to demonstrate the dynamics.