Abstract:
It is well-known that solutions of chemotaxis systems develop spiky structures and may even blow up in finite time. It is therefore a challenging task to develop numerical methods that are capable of accurately capture such type of solutions. Besides resolving the spikes, one has to make sure that the computed solution remains positive at all times since appearance of even small negative values of the cell densities typically leads to numerical "blow down", that is, to development of large negative structures which are totally unphysical and thus irrelevant.
I will first present several positivity preserving high-order hybrid finite-volume-finite-difference methods based on a uniform Cartesian grid. Even though these methods are quite accurate and robust, there are certain situations when one has to use very fine (and often impractical) mesh to achieve high resolution. We therefore develop an adaptive moving-mesh (AMM) strategy, which automatically shift most of the mesh point to the spiky parts of the solution. Our numerical experiments demonstrate that the second-order AMM method clearly outperforms the fourth-order method based on a uniform Cartesian mesh |