Talk ‘Structure-preserving numerical methods for nonlinear dispersive wave equations’ on 2024-02-15

I was invited to give a talk Structure-preserving numerical methods for nonlinear dispersive wave equations at the Séminaire de Calcul Scientifique et Modélisation at Université de Bordeaux (France) on Thursday, 2024-02-15, 14:00 CET.

Abstract

The numerical simulation of tsunami propagation is often based on the classical shallow water equations. However, there are several regimes where the assumptions used to derive this model are not satisfied. In this case, 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. The numerical methods will use the framework of the method of lines. Thus, we will discuss both spatial semidiscretizations and time integration methods. 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.