Kernel-based hybrid methods for convection-dominated equations on surfaces

By coupling appropriate high-resolution techniques, we aim to develop meshless numer- ical methods for solving partial differential equations (PDEs) on smooth surfaces whose solutions may contain shocks and/or discontinuities. We focus on both viscous and in- viscid scalar hyperbolic conservation laws and study second-order quasilinear diffusion- convection PDEs with zero or very slow diffusive transport, i.e., large P ́eclet number, as in many engineering applications. Both Meshless method of lines and various types of elliptic PDEs solvers (with finite difference discretization in time) are known to work well on PDEs with moderate P ́eclet number. Meshless semi-Lagrangian methods also show promising in solving pure advection PDEs on surfaces. This project begins with a literature review on existing meshless PDEs solvers. We are interested in numerical performance when solving PDEs on open surfaces with boundary conditions in order to identify good candidates to be used on smooth region. Note that all available error esti- mates and convergence analysis for (some kernel-based) surface PDE solvers only apply to closed surfaces. On the theoretical front, we focus on the meshless semi-Lagrangian methods in order to ensure that we use (domain type) integrator (on surface) correctly. The second part of this project will focus on the development time-dependent embedding methods. We conjecture that there exists some extensions to the velocity field in the advection and/or diffusion-convection PDEs so that the constant-along-normal solutions solves the corresponding embedded PDEs. After completing this analytical task, the embedded PDEs are readily solvable with ENO/WENO-type shock capturing algorithms in literature. As a proof of concept, the preliminary two dimensional results presented in this proposal show promising results in capturing shock and discontinuity in solution. The final task is to assemble a hybrid algorithm that run meshless and ENO/WENO solvers on different parts of the surface adaptively based on the regularities of the nu- merical solutions. The project will be concluded with numerical simulations of various benchmark PDEs on surfaces.