An implicit Euler finite-volume scheme for general cross-diffusion systems with volume-filling constraints is proposed and analyzed. The diffusion matrix may be nonsymmetric and not positive semidefinite, but the diffusion system is assumed to possess a formal gradient-flow structure that yields $L^\infty $ bounds on the continuous level. Examples include the Maxwell–Stefan systems for gas mixtures, tumor-growth models and systems for the fabrication of thin-film solar cells. The proposed numerical scheme preserves the structure of the continuous equations, namely the entropy dissipation inequality as well as the non-negativity of the concentrations and the volume-filling constraints. The discrete entropy structure is a consequence of a new vector-valued discrete chain rule. The existence of discrete solutions, their positivity, and the convergence of the scheme is proved. The numerical scheme is implemented for a one-dimensional Maxwell–Stefan model and a two-dimensional thin-film solar cell system. It is illustrated that the convergence rate in space is of order two and the discrete relative entropy decays exponentially.

中文翻译：

---

交叉扩散系统有限体积逼近的离散有界熵方法

提出并分析了具有体积填充约束的一般交叉扩散系统的隐式欧拉有限体积格式。扩散矩阵可能是非对称的并且不是半正定的，但假设扩散系统具有正式的梯度流结构，该结构在连续水平上产生 $L^\infty $ 边界。示例包括用于气体混合物的 Maxwell-Stefan 系统、肿瘤生长模型和用于制造薄膜太阳能电池的系统。所提出的数值方案保留了连续方程的结构，即熵耗散不等式以及浓度的非负性和体积填充约束。离散熵结构是一种新的向量值离散链式法则的结果。离散解的存在，它们的积极性，并证明了该方案的收敛性。该数值方案适用于一维 Maxwell-Stefan 模型和二维薄膜太阳能电池系统。说明空间中的收敛速度是二阶的，离散的相对熵呈指数衰减。