A new h-r hybrid moving mesh - level set method

In this thesis, we construct, implement, and discuss a new moving mesh framework, suitable for solving arbitrary time-dependent partial differential equations. Whereas r-refinement methods evolve a mesh smoothly, but are constrained to a fixed number of mesh nodes, and whereas h-refinement methods add or remove mesh nodes locally in a non smooth fashion, we introduce a new moving mesh framework, the h-r hybrid method, which is particularly useful for solutions where fine scale structures develop or dissipate. Essentially, a mesh is represented implicitly using level set functions. A moving mesh method is derived to update the level set function, implicitly updating the mesh; the level set function is updated such that a specified monitor function is equi-distributed by the implicitly represented grid nodes. Our construction allows for the natural creation or removal of mesh nodes at the domain boundaries; these mesh nodes are added or removed in a smooth fashion, consistent with {\em r}-adaptive schemes. Additionally, our hybrid method removes instabilities in current moving mesh methods by circumventing mesh crossing events. Following a review of moving mesh methods, level set ideas and related concepts, the new hybrid method is introduced. Various numerical examples are presented to validate our method in R^1 and R^2.