Talk ‘Structure-preserving numerical methods for dispersive wave equations’ at the Rhein-Main-Arbeitskreis ‘Mathematics of Computation’

I will give a talk Structure-preserving numerical methods for dispersive wave equations at the Rhein-Main-Arbeitskreis Mathematics of Computation on Friday, January 24, 2025, 16:15 - 17:00 in room 05-514, Staudingerweg 9 (Hilbert-Raum) in Mainz.

Abstract:

Several water wave propagation problems can be modeled using a depth-averaged shallow water approximation, e.g., tsunami propagation or dam breaks. In many cases, the classical first-order hyperbolic shallow water equations are sufficient to describe the wave dynamics. However, in some cases, higher-order effects need to be taken into account, leading to nonlinear dispersive wave equations. Several variants of such models exist and are used in practice. In this talk, we will review some recent developments of structure-preserving numerical methods. In particular, we will consider invariants such as the total energy and study efficient numerical methods yielding qualitative and quantitative improvements compared to standard schemes. To develop structure-preserving schemes, we make use of the general framework of summation-by-parts (SBP) operators in space, unifying the analysis of finite difference, finite volume, finite element, discontinuous Galerkin, and spectral methods. Finally, we combine structure-preserving spatial discretizations with relaxation methods in time to obtain fully-discrete, energy-conservative schemes.

Please find more information on the seminar series on the website of the Rhein-Main-Arbeitskreis Mathematics of Computation.