This is a past event.

Speaker: Professor Yang Yang, MTU;

Abstract: 

In this talk, we will apply the local discontinuous Galerkin methods to solve the classical Keller-Segel (KS) chemotaxis model. Chemotaxis is the highly nonlinear terminology which indicates movements by cells in reaction to a chemical substance, where cells approach chemically favorable environments and avoid unpleasant ones. Moreover, the model exhibits blow-up patterns with certain initial conditions. Biologically, finite-time blow up for solutions is expected to describe chemotactic collapse, that is the tendency of cells to concentrate to form spora, which can be explained mathematically as concentration towards a Dirac mass in finite time. We will give optimal rate of convergence under special finite element spaces before the blow-up occurs. To construct physically relevant numerical approximations, we consider P1-LDG scheme and develop a positivity-preserving limiter to the scheme. With this limiter, we can prove the L1-stability of the numerical scheme. Moreover, we will construct a special way to compute the numerical blow-up time and prove the convergence under mesh refinement. Finally, another energy stable LDG scheme will also be constructed.