Showing papers by "Nils Henrik Risebro published in 2008"
TL;DR: In this paper, a finite difference scheme for the Camassa-Holm equation is proposed, which can handle general $H^1$ initial data and converges strongly in $H 1$ towards a dissipative weak solution.
Abstract: We suggest a finite dfference scheme for the Camassa-Holm equation that can handle general $H^1$ initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of the Camassa-Holm equation.
48 citations
Posted Content•
TL;DR: In this article, an explicit finite difference scheme for the Camassa-Holm shallow water equation was proposed, which can handle general $H^1$ initial data and thus peakon-antipeakon interactions.
Abstract: 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.
13 citations
Journal Article•
TL;DR: In this article, an explicit finite difference scheme for the Camassa-Holm shallow water equation was proposed, which can handle general $H^1$ initial data and thus peakon-antipeakon interactions.
Abstract: 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 the Camassa-Holm equation.
11 citations
TL;DR: Simple and efficient finite-volume schemes of the Godunov type are devised based on a local decoupling of the system into a series of single conservation laws with discontinuous coefficients, hence termed semi-Godunov schemes.
Abstract: General m × m triangular systems of conservation laws in one space dimension are considered. These systems arise in applications like multi-phase flows in porous media and are non-strictly hyperbolic. Simple and efficient finite-volume schemes of the Godunov type are devised. These are based on a local decoupling of the sys- tem into a series of single conservation laws with discontinuous coefficients and are hence termed semi-Godunov schemes. These schemes are not based on the characteristic structure of the system. Some useful properties of the schemes are derived and several numerical experiments demonstrate their robustness and computational efficiency.
8 citations
TL;DR: In this article, a large-time-stepping method for conservation laws with source terms is presented, which is based on a local reformulation of the balance law as a conservation law with a discontinuous flux function.
Abstract: A well-balanced, large-time-stepping method for conservation laws with source terms is presented. The numerical method is based on a local reformulation of the balance law as a conservation law with a discontinuous flux function, and the approximate solution of this equation by a front tracking method. This yields an unconditionally stable method which is particularly well suited to calculate stationary states. The viability of this approach is demonstrated by several numerical examples.
5 citations