Efficient stochastic asymptotic-preserving scheme for tumor growth models with uncertain parameters

Abstract

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 behaviors 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.