A novel numerical strategy is introduced for computing approximations of solutions to a Cahn-Hilliard model with degenerate mobilities. This model has recently been introduced as a second-order phase-field approximation for surface diffusion flows. Its numerical discretization is challenging due to the degeneracy of the mobilities, which generally requires an implicit treatment to avoid stability issues at the price of increased complexity costs. To mitigate this drawback, we consider new first- and second-order Scalar Auxiliary Variable (SAV) schemes that, differently from existing approaches, focus on the relaxation of the mobility, rather than the Cahn-Hilliard energy. These schemes are introduced and analysed theoretically in the general context of gradient flows and then specialised for the Cahn-Hilliard equation with mobilities. Various numerical experiments are conducted to highlight the advantages of these new schemes in terms of accuracy, effectiveness and computational cost.

翻译：本文提出了一种新型数值策略，用于计算含退化迁移率的Cahn-Hilliard模型解的近似值。该模型近期被引入作为表面扩散流动的二阶相场近似。迁移率的退化特性对其数值离散构成挑战，通常需采用隐式处理以避免稳定性问题，但这会以增加计算复杂度为代价。为缓解这一缺陷，我们提出了新的标量辅助变量（SAV）一阶和二阶格式，与现有方法不同，这些格式聚焦于迁移率的松弛而非Cahn-Hilliard能量。本文在梯度流的通用框架下对这些格式进行了理论分析与引入，并针对含迁移率的Cahn-Hilliard方程进行了专门化处理。通过多项数值实验，从精度、有效性及计算成本等方面展示了新格式的优势。