| Description |
In this dissertation, we focus on the design, analysis, and implementation of robust, efficient, and highly accurate numerical algorithms based on Difference Potentials Method (DPM), with applications to the two-dimensional (2D) interface problems and the three-dimensional (3D) chemotaxis models in biology. We first develop the high-order accurate DPM-based algorithms for the 2D parabolic interface problems within a stationary composite domain. The prevalence of interface problems in the applications of modern science and engineering (e.g., multiphase fluid flow, composite anisotropic materials, etc.) and the lack of analytical tools, promote the employment of numerical algorithms. The primary challenge is to accurately and efficiently capture the jumps of the solutions or fluxes across the arbitrarily-shaped interface. Albeit there have been various numerical methods studied in the literature, we design novel and versatile DPM-based algorithms with high-order accuracy. The proposed methods can (i) handle general interface conditions; (ii) offer compatibility to any explicitly/implicitly-defined interface; and (iii) allow fast linear solvers based on uniform meshes. Moreover, we compare the DPM-based methods with two other state-of-the-art methods, i.e., the immersed cut-FEM and the conforming SBP-SAT-FD method. Next, we advance to the numerical simulations of the 3D chemotaxis models in the DPM framework. Chemotaxis describes the directed motion of organisms in response to chemical stimulus. Due to the analytical intractability of such models, numerical simulations are essential alternatives. However, there are multiple difficulties in designing numerical schemes for the 3D chemotactic processes. For example, the blow-up that represents the unbounded aggregations of cells, and the positivity property of the solutions to the models, especially in complex geometry, are numerically challenging to resolve. Hence, we propose efficient and accurate numerical algorithms based on DPM to handle various challenges in approximating chemotaxis systems. Furthermore, to enhance the computational efficiency, we design an unconventional domain decomposition approach, by extending the DPM-based algorithms for the 2D parabolic interface problems to the 3D chemotaxis models. The proposed domain decomposition method allows mesh adaptivity without loss of global accuracy, use of fast solvers, and easy parallelization of the algorithms in space. |
