We present and analyze a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time stepping scheme. The proposed method makes use of the underlying entropy structure of the system, expressing the main unknown in terms of the entropy variable by means of a nonlinear transformation. Such a transformation allows for imposing the physical positivity or boundedness constraints on the approximate solution in a strong sense. Moreover, nonlinearities do not appear explicitly within differential operators or interface terms in the scheme, which significantly improves its efficiency and ease its implementation. We prove the existence of discrete solutions and their asymptotic convergence to a weak solution to the continuous problem. Numerical results for some one- and two-dimensional problems illustrate the accuracy and entropy stability of the proposed method.

翻译：本文提出并分析了一种用于近似求解非线性交叉扩散系统的结构保持方法，该方法将局部间断伽辽金空间离散化与后向欧拉时间步进格式相结合。所提方法利用了系统的内在熵结构，通过非线性变换将主要未知量用熵变量表示。这种变换能够以强形式对近似解施加物理上的正定性或有界性约束。此外，非线性项不会显式出现在格式的微分算子或界面项中，这显著提高了计算效率并简化了实施过程。我们证明了离散解的存在性及其向连续问题弱解的渐近收敛性。针对若干一维和二维问题的数值结果验证了所提方法的精度与熵稳定性。