Numerical Methods for Wave Propagation
In this course, we discussed the fundamentals of Finite Volume and discontinuous Galerkin methods for the discretization of hyperbolic equations, focusing on the shallow water equations, as they are used in tsunami and storm surge simulation. The concept of numerical fluxes across element interfaces was introduced for linear and non-linear (e.g. shallow water waves) hyperbolic systems using the eigenstructure of the equation system.
For discontinuous Galerkin methods, the lectures focused on the choice of the polynomial basis functions, element types (esp. for triangular elements), limiters, the particular form of the mass, flux, differentiation, and stiffness matrices. We also described high-order Taylor expansion for time integration including explicit, semi-implicit, and fully-implicit methods. Special topics, such as local adaptation of the approximation order or the local time-stepping technique were also briefly discussed.
Overview on the lecture sessions:
- Finite Volume I: theory of hyperbolic problems in 1D, Riemann problems (R.J. LeVeque)
- Finite Volume II: Godunov' method, high-resolution methods (limiters), 2D methods (R.J. LeVeque)
- Finite Volume III: shallow water equations with topography, approximate Riemann solvers, wave propagation method (R.J. LeVeque)
- Introduction to Discontinuous Galerkin methods (F.X. Giraldo)
- 1D application of DG methods (F.X. Giraldo)
- 2D application of DG methods (F.X. Giraldo)
- Conforming triangular mesh generation and efficiency (J. Behrens)
- Adaptive mesh refinement for non-conforming DG methods (M.A. Kopera, F.X. Giraldo)
- Development of a coastal inundation model using a triangular DG method (S. Gopalakrishnan, F.X. Giraldo)
The course was accompanied by tutorials based on the GeoClaw package (R.J. LeVeque) and the DGCOM code (F.X. Giraldo).
See the following website for course material on the Finite Volume lectures and tutorials by R.J. LeVeque: