Discontinuous Galerkin models

of shallow-water phenomena

by Dr Timothy Warburton

Department of Computational and Applied Mathematics, Rice University


(1) Background machinery for Discontinuous Galerkin methods

  • Orthogonal polynomials
  • Langrange polynomials and interpolation
  • Differentiation matrices
  • Projection operators
  • Lift matrices
  • Cubature: multi-dimensional quadrature rules and dealiasing

(2) Discretization of Discontinuous Galerkin methods for the shallow water equations

  • Discretization of the Shallow Water equations using DG methods on triangular elements, with emphasis on unstructured/adaptive grids
  • Wetting-drying algorithms for run-up simulations

(3) Advanced Time-Integration for the Shallow Water Equations

  • Method of lines
  • High-order explicit methods
  • Implicit-explicit methods (including multi-step single-stage and single-step multi-stage methods)
  • Fully-implicit methods
  • Multi-rate methods

(4) CUDA & OpenCL kernels for DG shallow water and tsunami simulations

  • CUDA & OpenCL kernels for time-stepping and for computing the volume and flux integrals in DG methods
  • Setting up the initial grid of any region of the world oceans
  • Bathymetry
  • Initial conditions
  • Role of data assimilation/observability of such events
  • Loo.py based kernel generation