Nils Henrik Risebro

Boris Andreianov, Kenneth H. Karlsen, Nils Henrik Risebro

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$Γ$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

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.

Giuseppe Maria Coclite, Kenneth H. Karlsen, Nils Henrik Risebro

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation.

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.

Håkon Hoel, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro et al.

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with a rough path dependent flux function. For a convex flux, it is demonstrated that rough path oscillations may lead to "cancellations" in the solution. Making use of this property, we show that for $α$-H{ö}lder continuous rough paths the convergence rate of the numerical methods can improve from $\mathcal{O}(\text{COST}^{-γ})$, for some $γ\in \left[α/(12-8α), α/(10-6α)\right]$, with $α\in (0, 1)$, to $\mathcal{O}(\text{COST}^{-\min(1/4,α/2)})$. Numerical examples support the theoretical results.

Neelabja Chatterjee, Nils Henrik Risebro

We study the periodic Ostrovsky-Hunter equation in the case where the flux function may depend on the spatial variable. Our main results are that if the flux function is twice differentiable, then there exists a unique entropy solution. This entropy solution may be constructed as a limit of approximate solutions generated by a finite volume scheme, and the finite volume approximations converge to the entropy solution at a rate 1/2.