Numerical Schemes for Conservation Laws via Hamilton-Jacobi Equations

Abstract

Abstract. We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik's entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).

On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations

“…This section is devoted to the construction and the convergence analysis of a semi-lagrangian scheme for the time discretization of system (1.1)-(1.2) over the time interval [0, T ]. For an introduction to this class of schemes we refer to [1, Appendix B] and [8,9,10,11]. To proceed, we need to refine the previous hypothesis on the hamiltonian H and to add assumptions on the growth of H w.r.t.…”

Convergence results for an inhomogeneous system arising in various high frequency approximations

“…If one looks for planar wave solutionsũ(t, x) = u(x) exp(ikt) to the wave equation (9), one is led to find a steady function u satisfying the scalar Helmholtz equation:…”

“…loc (]0, T [×R) to a = ∂ x ϕ (see Theorems 1.1 and 2.2 in [9]). Next, the strong CFL condition (38) is trivially satisfied by the Hamiltonian p 2 /2, so a discrete semiconcavity estimate holds.…”

Initial data identification in conservation laws and Hamilton–Jacobi equations

“…Geometric and topological properties of this set are also obtained. To this aim, a refinement of the results in [22], see also [12,23], on the relation between (1.1) and (1.2) had to be obtained.…”