Showing papers by "Nils Henrik Risebro published in 2007"

Numerical schemes for computing discontinuous solutions of the Degasperis–Procesi equation

Abstract: Recent work (COCLITE, G. M. & KARLSEN, K. H. (2006) On the well-posedness of the Degasperis-Procesi equation. J. Funct. Anal., 233, 60-91) has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper, we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrability, peakon solutions and H 1 as the relevant functional space).

Viscosity solutions of hamilton–jacobi equations with discontinuous coefficients

Abstract: We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

Convergent difference schemes for the Hunter–Saxton equation

Abstract: We propose and analyze several finite difference schemes for the Hunter-Saxton equation (HS) u t + uu x = 1 2 ∫ x 0 (u x ) 2 d x , x > 0, t > 0. This equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.

A convergent finite difference method for a nonlinear variational wave equation

Abstract: We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$ with $u|_{t=0}=u_0$ and $u_t|_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c u_x$ and $S=u_t-c u_x$, the variational wave equation is equivalent to $R_t-c R_x=\tilde c (R^2-S^2)$ and $S_t+c S_x=-\tilde c (R^2-S^2)$ with $\tilde c=c'/(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R|_{t=0}, S|_{t=0}\in L^1\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples.