Numerical methods for the stochastic wave equation and homogenization for Li-ion batteries

Abstract

The first part of this thesis deals with the numerical approximation of the linear and the semi-linear stochastic wave equation, where the main objective is to study strong convergence of the fully discrete problem. More specifically, we apply the discontinuous Galerkin (dG) finite element method in space and take the semigroup approach to derive optimal strong convergence rates. For the temporal approximation we use a stochastic extension of the explicit position Verlet method. The stability and strong convergence of our scheme is studied under a CFL condition. Further, we analyze the energy conservation of the considered numerical approximations. The numerical investigations verify the theoretical results of the numerical discretizations. Our numerical experiments show that the stochastic position Verlet scheme is the most efficient amongst other time integrators despite the CFL restriction. The second part of this thesis is concerned with the mathematical modeling of lithium-ion batteries (Li-batteries). The main aim of this part is to derive an effective macroscopic description of active electrodes. We first state governing equations describing the transport of ions and electrons in a Li-battery consisting of an anode, a polymer electrolyte as separator, and a composite cathode composed of electrolyte and a single intercalation particle and extend this transport formulation towards heterogeneous cathodes. The reaction rates at the solid-electrolyte interface and at the anode are modeled using the Butler-Volmer equations. Based on the microscopic formulation accounting for ion and electron transport in composite cathodes, we derive homogenized charge transport equations via the method of asymptotic two-scale expansions from homogenization theory.