Helge Holden

Helge Holden, Ujjwal Koley, Nils Henrik Risebro

We prove convergence of a fully discrete finite difference scheme for the Korteweg--de Vries equation. Both the decaying case on the full line and the periodic case are considered. If the initial data $u|_{t=0}=u_0$ is of high regularity, $u_0\in H^3(\R)$, the scheme is shown to converge to a classical solution, and if the regularity of the initial data is smaller, $u_0\in L^2(\R)$, then the scheme converges strongly in $L^2(0,T;L^2_{\mathrm{loc}}(\R))$ to a weak solution.

Rajib Dutta, Helge Holden, Ujjwal Koley et al.

In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data are sufficiently regular, fully discrete finite difference schemes shown to converge to a classical solution. Finally, the convergence is illustrated by several examples.

Helge Holden, Kenneth H. Karlsen, Trygve K. Karper

We analyze splitting algorithms for a class of two-dimensional fluid equations, which includes the incompressible Navier-Stokes equations and the surface quasi-geostrophic equation. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data are sufficiently regular.

Helge Holden, Kenneth H. Karlsen, Trygve K. Karper

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar unknown itself and hence the equations are so-called active scalar equations. Examples are provided by the surface quasi-geostrophic and aggregation equations. In addition, Burgers-type equations with fractional diffusion as well as the KdV and Kawahara equations are covered. Our main result is that the Godunov and Strang splitting methods converge with the expected rates provided the initial data is sufficiently regular.