求解非线性交叉扩散问题的快速稳定finite difference格式
Fast stable finite difference schemes for nonlinear cross-diffusion
摘要：
交叉扩散模型的动态特性导致隐式差分格式的高计算复杂性，使其不适合需要实时结果的任务。为了降低计算量，我们提出了两个算子分裂方案用于非线性交叉扩散过程，并使用离散$L^2$能量方法建立其稳定性。此外，通过实现一个稳定的分解系统矩阵作为一个向前-向后通过，对应于托马斯算法的自扩散过程，我们表明，使用隐式交叉扩散可以在执行时间方面的竞争力，拓宽了可行的交叉扩散系数的范围\textit{on-the-fly}应用。
英文摘要：
The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete $L^2$ energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for \textit{on-the-fly} applications.
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