In the first part, we introduce a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we show that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. In the second part, we propose an accurate front capturing scheme for a class of tumor growth models with a free boundary limit. We show that the semi-discrete scheme naturally connects to the free boundary limit equation. With proper spatial discretization, the fully discrete scheme has improved stability, preserves positivity, and can be implemented without nonlinear solvers.

Zhennan Zhou is an assistant professor at BICMR, Peking University since 2017. He was a William W. Elliott Assistant Research Professor at Duke University, from 2014 to 2017. He obtained his PhD degree from University of Wisconsin-Madison in 2014. His primary research interests are in the applied analysis of PDE and stochastic models, and numerical approximation of scientific problems arising from quantum mechanics, theoretical chemistry, solid state physics, materials science, biology, etc.