The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting and segmentation.

中文翻译：

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有界质量源 Cahn-Hilliard 系统松弛标量辅助变量格式的稳定性和收敛性

Shen 等人的标量辅助变量（SAV）方法。（2018）提出了一种离散一大类梯度流的新颖方法，许多作者对一般耗散系统进行了扩展和改进。在这项工作中，我们考虑具有质量源的卡恩-希利亚德系统，对于图像处理和生物应用，可能不允许涉及金兹堡-朗道能量的耗散结构。因此，与之前的工作相比，此类系统的 SAV 离散解决方案的稳定性并不是立竿见影的。我们在有界质量源的情况下，为 Jiang 等人意义上的一阶松弛 SAV 方案建立了时间离散解的稳定性和收敛性。（2022），并将我们的想法应用到 Cahn-Hilliard 系统中，质量源出现在二嵌段共聚物相分离、肿瘤生长、图像修复和分割中。