In this paper, we investigate a class of tumor growth models governed by porous medium-type equations with uncertainties arisen from the growth function, initial condition, tumor support radius or other parameters in the model. We develop a stochastic asymptotic preservation (s-AP) scheme in the generalized polynomial chaos-stochastic Galerkin (gPC-SG) framework, which remains robust for all index parameters $m\geq 2$. The regularity of the solution to porous medium equations in the random space is studied, and we show the s-AP property, ensuring the convergence of SG system on the continuous level to that of Hele-Shaw dynamics as $m \to \infty$. Our numerical experiments, including capturing the behaviours such as finger-like projection, proliferating, quiescent and dead cell's evolution, validate the accuracy and efficiency of our designed scheme. The numerical results can describe the impact of stochastic parameters on tumor interface evolutions and pattern formations.

翻译：本文研究了一类由多孔介质型方程控制的肿瘤生长模型，其不确定性来源于生长函数、初始条件、肿瘤支撑半径或模型中的其他参数。我们在广义多项式混沌-随机伽辽金框架下，发展了一种随机渐近保持格式，该格式对所有指标参数$m\geq 2$均保持鲁棒性。研究了多孔介质方程解在随机空间中的正则性，并证明了该格式的随机渐近保持特性，确保了当$m \to \infty$时，连续层面上的随机伽辽金系统收敛于Hele-Shaw动力学系统。我们的数值实验，包括捕捉指状突起、增殖细胞、静止细胞与死亡细胞的演化等行为，验证了所设计格式的精确性与高效性。数值结果能够描述随机参数对肿瘤界面演化与模式形成的影响。