PhD comprehensive exam | Andrew Giuliani, Efficient mesh adaptivity for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on graphics processing units

Efficient mesh adaptivity for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on graphics processing units

Abstract

In the natural sciences, many physical phenomena such as fluid behaviour can be described with conserved quantities, e.g. mass, momentum and energy.  Hyperbolic conservation laws are a type of partial differential equation (PDE) that result from this conservation property. The Euler equations of gas dynamics are one notable example; they can be used to describe the flow around an airfoil or aircraft. Because the solutions of these equations may exhibit shocks or discontinuities, a numerical method that can accurately represent such features is needed. The discontinuous Galerkin (DG) method is a high-order method without an extensive stencil that can successfully describe shocks. In the Scientific Computing research group of Applied Mathematics, we have software implementing the DG method on NVIDIA GPUs in CUDA C.

This talk will focus on an efficient h-adaptivity algorithm whereby the mesh is spatially refined or coarsening according to a refinement indicator.  Typically, data structures such as trees are required in order to maintain mesh connectivity; the implementation of this on GPUs is subtle and must be approached with care.  Other key aspects of our h-adaptive DG-GPU implementation will also be discussed.  Finally, future avenues of research will presented such as p-enrichment, adaptivity indicators and local timestepping.