Convergence of a variational time discretization for the 1-dimensional isentropic Euler equations

Nguyen, Trong-Hieu; Westdickenberg, Michael (Thesis advisor); Müller, Siegfried (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

This dissertation is devoted to the proof of convergence of a variational time discretization proposed in [1]. Starting from the variational time discretization, we apply the well-known compensated compactness framework to prove strong convergence of constructed solutions to the p-system instead of the isentropic Euler equations. The reason behind this alteration is that the variational time discretization has a Lagrangian nature: Fluid states are constructed based on a transport map, which is obtained by solving a minimization problem at each time step. These transport maps update the position of all fluid particles after each iteration, thus illustrate their trajectories. Once we obtain the strong convergence for the p-system, the interchangeability between the two systems following from continuity arguments will lead to the desired strong convergence of the scheme for isentropic Euler equations.