High order shock capturing methods with compact stencils for use with adaptive mesh refinement and mapped grids
This thesis focuses on several developments toward creating a high order shock capturing method that can be used on mapped grids with block-structured adaptive mesh refinement (AMR). The discontinuous Galerkin (DG) method is used as a starting point for the construction of this method. A high order mapped grid DG scheme is implemented and tested on several hyperbolic PDEs. It is shown that even on highly-skewed meshes these DG schemes can illustrate high order convergence. Additionally a family of limiters is developed that is extremely flexible with respect to geometry. This flexibility originates from the fact that these limiters use a minimal stencil and do not require directional information. The performance of this family of limiters is explored on structured, unstructured and mapped grids. Lax-Wendroff time stepping schemes have a very compact stencil and they can easily be used with local time stepping because they produce a local space-time Taylor series of the solution. A positivity limiter is developed to allow the use of high order Lax-Wendroff time stepping on PDEs, such as the Euler equations, that require the positivity of pressure and density. Additionally a new type of Lax-Wendroff time stepping, known as the differential transform method, is adapted to both a WENO finite difference method and DG. The differential transform method uses tools from the automatic differentiation literature to automate the computation of space-time Taylor series. A high order DG scheme using the differential transform method is developed to use block-structured AMR and local time stepping. This method is implemented in one dimension and found to be very effective at maintaining the accuracy of the high order DG method while reducing its computational cost. The accuracy and convergence rates of the methods developed in this thesis are established by comparing to analytical or very highly refined numerical solutions. All of the methods developed, with the exception of the positivity limiter, are tested on the advection equations, the acoustic equations and the Euler equations on a variety of standard test problems found in the literature.
- Applied mathematics