Front capturing schemes for nonlinear PDEs with a free boundary limit

Abstract: Evolution in physical or biological systems often involves interplay between nonlinear interaction among the constituent “particles”, and convective or diffusive transport, which is driven by density dependent pressure. When pressure-density relationship becomes highly nonlinear, the evolution equation converges to a free boundary problem as a stiff limit. In terms of numerics, the nonlinearity and degeneracy bring great challenges, and there is lack of standard mechanism to capture the propagation of the front in the limit. In this talk, I will introduce a numerical scheme for tumor growth models based on a prediction-correction reformulation, which naturally connects to the free boundary problem in the discrete sense. As an alternative, I will present a variational method for a class of continuity equations (such as Keller-Segel model) using the gradient flow structure, which has built-in stability, positivity preservation and energy decreasing property, and serves as a good candidate in capturing the stiff pressure limit.