DMS Graduate Student Seminar

Speaker: Lianzhang Bao

Title: Numerical study of chemotaxis system with a moving boundary

Abstract: Chemotaxis is a fundamental sensory phenomenon by which biological cells translate environmental chemical information into motile behavior. In this talk, I will first introduce and review several chemotaxis models including the chemotaxis system with a moving boundary defined by Stefan condition to describe the dynamics of the spreading population. Then I will focus on the numerical study of the logistic type chemotaxis with a moving boundary. There are several numerical difficulties to efficiently handle such systems. Firstly, the solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. Secondly, extremely small time steps are usually demanded due to the stiffness of the system. Thirdly, it is always difficult to efficiently and accurately handle the moving boundaries. To overcome them, a front fixing framework coupled with a finite difference and finite volume method is introduced. The accuracy of the proposed method, the positivity of the solution, and the stability of the scheme will be discussed. Some open problems and future work will also be presented.