Seminar Computational and Applied Mathematics

Abstract: This Master's thesis deals with the numerical approximation of linear hyperbolic problems appearing in rolling contact mechanics. First, the existence and uniqueness of strict solutions to the considered equations, which contain nonlocal and boundary terms, are analysed within the framework provided by the semigroup theory. Then, the space semi-discrete problem is formulated using the discontinuous Galerkin finite element method (DGMs), by replacing the unbounded operator appearing in the abstract formulation with a finite-dimensional one. Quasi-optimal error convergence is obtained for the space semi-discrete scheme by introducing upwind regularisation. Time discretisation is then achieved by relying on explicit first and second-order Runge-Kutta algorithms (RK1 and RK2, respectively), yielding quasi-optimal convergence in time owing to certain refined CFL conditions. In particular, the considered RK2 schemes cover the explicit midpoint method, Heun's second-order method, and Ralston's method.