I was invited to give a talk Efficient and robust numerical methods based on adaptivity and structure preservation at the Seminar of Computational and Numerical Mathematics at TU Hamburg-Harburg (TUHH) on Wednesday, 2023-07-05, 12:00 CEST.
We present some recent developments for the numerical simulation of transport-dominated problems such as compressible fluid flows and nonlinear dispersive wave equations. We begin with a brief review of modern entropy-stable semidiscretizations of hyperbolic conservation laws and use the method of lines to obtain efficient, fully discrete numerical methods. Next, we introduce means to preserve the entropy structures also under time discretization. Therefore, we present the relaxation approach, a recent technique developed as small modifications of standard time integration schemes such as Runge-Kutta or linear multistep methods, which is designed to preserve the conservation or dissipation of important functionals of the solution. This can be an entropy in the case of compressible fluid flows, the energy of Hamiltonian problems, or another nonlinear invariant.