Computational methods in pattern formation solutions

by Yongtao Zhang - (Notre Dame)

In this talk, I will present two kinds of numerical methods for mathematical models in biological pattern formation problems. The first method is the weighted essentially non-oscillatory (WENO) method for solving the nonlinear chemotaxis models. Chemotaxis is the phenomenon in which cells or organisms direct their movements according to certain gradients of chemicals in their environment. Chemotaxis plays an important role in many biological processes, such as bacterial aggregation, early vascular network formation, among others. While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this work, we combined two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO
reconstruction on complex mesh geometries. The second method is the Krylov implicit integration factor (IIF) method for nonlinear reaction–diffusion and advection-reaction-diffusion equations in pattern formations. Integration factor methods are a class of ‘‘exactly linear part’’ time discretization methods. Efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal
discretization was open. Here, I will present our results in solving this problem by applying the Krylov subspace approximations to the matrix exponential operator. We applied this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction–diffusion equations. Then we extended the Krylov IIF method to solve advection-reaction-diffusion PDEs and achieved high order accuracy. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the methods.