| Adaptive mesh refinement strategies are proposed for the immersed interface method. The immersed interface method (IIM) was developed to solve partial differential equations involving interfaces or discontinuities. The IIM modifies the standard finite difference schemes around the interface by using jump conditions to improve the accuracy. The adaptive mesh refinement (AMR) technique is to deploy a high mesh resolution only around the interface where it is most needed, whereas use a comparatively coarse resolution where the solution is smooth enough. The AMR utilizes the computational resources more economically without loss of accuracy, compared to the mesh refined uniformly. We developed an adaptive version of the IIM to take advantages of both: the numerical accuracy of the IIM and the computational efficiency of AMR. Moreover, we designed the AMR-IIM based on the popular level set method to solve many interface problems.;In this dissertation, we developed a brand new mesh generation method, coupled with an efficient data structure for the AMR-IIM. The local mesh refinement is generated from a narrow tube |ϕ(x, y, t)| ≤ delta around the interface. The linear system of equations derived from the finite difference discretization is solved by an algebraic multigrid solver. The AMR-IIM has been successfully applied to elliptic partial differential equations, Stokes equations, Navier-Stokes equations with fixed or moving interfaces. Numerical results show that the AMR-IIM inherits second order accuracy of the previous IIM on a uniform mesh, and runs faster since the size of linear system of equations is reduced by more than 50%. Implementation details and plenty of numerical experiments are also presented in the dissertation. |
