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 $L^\infty$ 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 $L^\infty$ boundedness. Furthermore, we will analyze the $L^2$ error and present some numerical simulations to demonstrate the dynamics.

翻译：相场模型是具有能量随时间自然耗散特性的梯度流。为保持该特性，已有大量数值格式被深入研究。本文研究一类著名方法——指数积分器方法（EI）。现有文献中，部分工作研究了针对不同相场模型的EI格式，并通过要求非线性源项满足强Lipschitz条件或数值解满足特定$L^\infty$有界性（最大值原理）来证明能量耗散。然而，对于（非局部）Cahn-Hilliard方程等相场模型，最大值原理不再成立。因此，如何通过EI格式求解该类模型长期悬而未决。本文以Cahn-Hilliard方程为对象，通过一阶EI格式证明能量耗散，系统性建立了EI型格式对该类模型的适用方法。二阶EI格式可类似处理，将在后续论文中讨论。据我们所知，这是首个不依赖强Lipschitz条件或$L^\infty$有界性处理相场模型的研究。此外，我们将分析$L^2$误差并进行数值模拟以展示动力学行为。