一个演化型交叉扩散癌症侵袭模型的通量校正输运稳定化
Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model
AI总结：
针对趋化项占优时交叉扩散癌症侵袭模型直接数值求解可能不稳定的问题，提出结合非线性通量校正输运的隐式有限元格式，证明其可解且保持正性，并通过二维实验验证性能。
摘要：
在本工作中，我们研究了一个癌细胞侵袭健康组织的模型，该模型由一个非线性偏微分方程组描述，包括一个交叉扩散-反应方程以及另外两个非线性常微分方程。我们表明，当该系统的对流部分（即趋化项）占主导地位时，针对所研究系统的直接数值方法可能不稳定。我们提出一种隐式有限元方法，使用协调的 $P_1$ 或 $Q_1$ 有限元对模型进行空间离散，并采用 $\ heta$ 方法进行时间离散。离散问题通过非线性通量校正输运方法进行稳定化。文中证明，非线性格式以及定点迭代中使用的线性化问题都是可解且保持正性的。我们使用 deal.II 库给出了若干二维数值实验，以展示所提方法的性能。
英文摘要：
In the present work, we investigate a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We show that when the convective part of the system, the chemotactic term, is dominant, then straightforward numerical methods for the studied system may be unstable. We present an implicit finite element method using conforming $P_1$ or $Q_1$ finite elements to discretize the model in space and the $θ$-method for discretization in time. The discrete problem is stabilized using a nonlinear flux-corrected transport approach. It is proved that both the nonlinear scheme and the linearized problems used in fixed-point iterations are solvable and positivity preserving. Several numerical experiments are presented in 2D using the deal.II library to demonstrate the performance of the proposed method.
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