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：本文提出了一种新颖的数值策略，用于计算含退化迁移率的Cahn-Hilliard模型解的逼近。该模型近期被引入作为表面扩散流的二阶相场近似。由于迁移率的退化性，其数值离散面临挑战，这通常需要隐式处理以避免稳定性问题，但代价是增加计算复杂度。为缓解这一不足，我们考虑了一阶和二阶的新型标量辅助变量（SAV）格式，与现有方法不同，这些格式侧重于迁移率的松弛而非Cahn-Hilliard能量。本文在梯度流的通用背景下对这些格式进行了理论引入与分析，随后将其专门应用于含迁移率的Cahn-Hilliard方程。我们开展了多项数值实验，以突出这些新格式在精度、有效性和计算成本方面的优势。