Showing papers by "Nils Henrik Risebro published in 2015"
01 Jan 2015
TL;DR: In this article, it was shown that the limit found by front tracking, that is, the weak solution of the initial value problem, is stable in L 1 with respect to perturbations in the initial data.
Abstract: The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem
$$\displaystyle u_{t}+f(u)_{x}=0,\quad u(x,0)=u_{0}(x),$$
(7.1)
is stable in L 1 with respect to perturbations in the initial data. In other words, if \(v=v(x,t)\) is another solution found by front tracking, then
$$\displaystyle{\left\|u(\,\cdot\,,t)-v(\,\cdot\,,t)\right\|}_{1}\leq C{\left\|u_{0}-v_{0}\right\|}_{1}$$
for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.
27 citations
TL;DR: In this article, a fully discrete finite difference scheme for the Korteweg-de Vries equation was shown to converge strongly in L 2(0,T;L2loc(R)) to a weak solution.
Abstract: 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=u0 is of high regularity, u0∈H3(R), the scheme is shown to converge to a classical solution, and if the regularity of the initial data is less, u0∈L2(R), then the scheme converges strongly in L2(0,T;L2loc(R)) to a weak solution.
26 citations
TL;DR: In this paper, the convergence of higher order schemes for the Cauchy problem associated with the KdV equation was studied and a Galerkin-type implicit scheme was proposed, which has higher order accuracy in space and first-order accuracy in time.
Abstract: We study the convergence of higher order schemes for the Cauchy problem associated with the KdV equation. More precisely, we design a Galerkin-type implicit scheme which has higher order accuracy in space and first order accuracy in time. The convergence is established for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0,T;L^2_{{loc}}(\mathbb{R}))$ to a weak solution. Finally, the convergence is illustrated by several numerical examples.
20 citations
TL;DR: In this paper, the convergence of both Godunov and Strang splittings was shown for the Benjamin-Ono equation, i.e., the initial data are sufficiently regular.
8 citations
01 Jan 2015
TL;DR: The analysis in one dimension can be carried over to higher dimensions by essentially treating each dimension separately, and the final results are very much the natural generalizations one would expect.
Abstract: Our analysis has so far been confined to scalar conservation laws in one dimension. Clearly, the multidimensional case is considerably more important. Luckily enough, the analysis in one dimension can be carried over to higher dimensions by essentially treating each dimension separately. This technique is called dimensional splitting. The final results are very much the natural generalizations one would expect.
1 citations