Solving Korteweg-de Vries equations with Discontinuous Galerkin methods

Abstract: In this thesis the Discontinuous Galerkin approximation performance applied to the Korteweg–de Vries equation is investigated. This equation is nonlinear with a third spatial derivative and can be used for shallow water movement. The thesis includes a background in numerical methods on conservation laws, Discontinuous Galerkin methods and the Korteweg-de Vries equation. To approximate the third order derivative, the thesis reformulates Korteweg-de Vries equation as a system of first order equations in order to apply Discontinuous Galerkin effectively. The thesis presents two choices of numerical fluxes, central- and alternating flux which both show promising convergence results, similar to that of the first order problem. Stability of the numerical approximation is proved analytically while convergence is shown numerically. The central flux appear to have spectral convergence, O(h^m) for even approximation order $m$ and sub-optimal convergence for $m$ odd while alternating flux shows spectral convergence for all approximation orders, $h$ being the discretization mesh. However, the central flux is found, in practice, to be only half as stiff and thus one should choose the numerical flux by the problem at hand.