Analysis and Implementation for an Time Euler Implicit - Space Finite ElementApproximation To a Hasegawa-Mima Plasma Model

Abstract

In Magnetohydrodynamics, the Hasegawa-Mima equation appears as a model for pseudo three-dimensional turbulence of a confined plasma inside a Tokamak reactor. This equation bears close resemblence to the 2d Navier-Stokes equation for an incompressible fluid. In this thesis, we apply an in time Euler implicit - space Finite Element Galerkin method to obtain solutions of this equation that satisfy periodic boundary values over the square. The Galerkin method readily provides us with a numerical scheme for which we prove convergence with minimal constraints on the time step, while making use of periodicity to obtain a priori esimates. Finally, we seek initial data for which the corresponding solution of the Hasegawa-Mima equation is a traveling wave. Those initial data turn out to be solutions to a semi-linear-elliptic periodic problem over a square, and we approximate them using a Newton-Galerkin method.