Course Program

Numerical Methods for Wave Propagation

In this course, we will introduce 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 is 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 will discuss the choice of the polynomial basis functions, element types (focusing on triangular elements), limiters, the particular form of the mass, flux, differentiation, and stiffness matrices. This course will also describe 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 will be discussed briefly, at least.

The course will thus be organized into the following lectures:

  1. Finite Volume methods for hyperbolic problems (fundamentals)
  2. Flux computation for linear and non-linear problems
  3. Finite Volume discretization of the shallow water equations
  4. Introduction to discontinuous Galerkin methods
  5. Discontinuous Galerkin methods for the shallow water equations
  6. Higher-order discontinuous Galerkin
  7. Adaptive discontinuous Galerkin methods
  8. Numerical issues resulting from inundation: wetting and drying

The course will be accompanied by tutorials, where students will implement basic FV and DG algorithms to understand the advantages of high-order approximation and the importance of the correct treatment of space- and time-dependent parts of the wave equations.