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 $\theta$-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.

翻译：本文研究了一个描述癌细胞侵入健康组织的模型，该模型由非线性偏微分方程组构成，包含一个交叉扩散-反应方程和两个额外的非线性常微分方程。我们证明，当系统的对流部分（趋化项）占主导地位时，直接采用常规数值方法求解该模型可能导致不稳定性。我们提出了一种隐式有限元方法，在空间离散上使用协调的$P_1$或$Q_1$有限元，在时间离散上采用$\theta$方法。通过非线性通量校正传输方法对离散问题进行稳定化处理。证明了非线性格式及不动点迭代中使用的线性化问题均具有可解性和正性保持特性。利用deal.II库在二维空间中进行了多项数值实验，展示了所提方法的性能。