Numerical studies of geometric partial differential equations with symplectic methods

Abstract

In this thesis the (2+1) dimensional wave map equations with the 2- sphere as target manifold is solved, using numerical methods. The focus will be on two different, but related topics. First, the numerical results obtained by using the standard fourth order Runge-Kutta scheme and the symplectic, constraint preserving Rattle method are compared. We will see that in terms of constraint, energy and symmetry preservation the Rattle method shows superior behaviour, especially in critical situations and long term evolutions. To investigate the long term behaviour, wave map equations with additional source terms are used for the first time.

In the second part, the blow-up dynamics and the singularity formation of the (2+1) dimensional wave map system will be studied. The results from simulations of the equivariant case in the first and second homotopy class are presented. The latter are novel results and have not been done anywhere before. The results from the first homotopy class show a similar behaviour to the results, already known in the literature. Finally, the system is extended to the non-equivariant case and study the critical behaviour there. We are able to observe a similar behaviour as in the equivariant case. In addition numerical evidence is presented that the equivariant blow-up scenario is stable under non-equivariant perturbations.

Before numerical results and their analysis are presented, we give an overview of the theoretical background of the equations as well as detailed descriptions of the numerical methods we used.