25 research outputs found
The design of conservative finite element discretisations for the vectorial modified KdV equation
We design a consistent Galerkin scheme for the approximation of the vectorial
modified Korteweg-de Vries equation. We demonstrate that the scheme conserves
energy up to machine precision. In this sense the method is consistent with the
energy balance of the continuous system. This energy balance ensures there is
no numerical dissipation allowing for extremely accurate long time simulations
free from numerical artifacts. Various numerical experiments are shown
demonstrating the asymptotic convergence of the method with respect to the
discretisation parameters. Some simulations are also presented that correctly
capture the unusual interactions between solitons in the vectorial setting
Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations
We develop and analyze the first hybridizable discontinuous Galerkin (HDG)
method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show
that the semi-discrete scheme is stable with proper choices of the
stabilization functions in the numerical traces. For the linearized fifth-order
equations, we prove that the approximations to the exact solution and its four
spatial derivatives as well as its time derivative all have optimal convergence
rates. The numerical experiments, demonstrating optimal convergence rates for
both the linear and nonlinear equations, validate our theoretical findings
Energy consistent DG methods for the Navier-Stokes-Korteweg system
We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems