Stability of reconstruction schemes for scalar hyperbolic conservations laws

Abstract

Abstract. We study the numerical approximation of scalar conservation laws in dimension 1 via general reconstruction schemes within the finite volume framework. We exhibit a new stability condition, derived from an analysis of the spatial convolutions of entropy solutions with characteristic functions of intervals. We then propose a criterion that ensures the existence of some numerical entropy fluxes. The consequence is the convergence of the approximate solution to the unique entropy solution of the considered equation.

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

“…We see that the more precise information we have brought on each cell C j for calculating the numerical fluxes makes the scheme less diffusive than the original one. This strategy was proposed (and further discussed in detail) in [21,22] (see also [12,20]). In particular, it is shown therein that the numerical solution presented in Figure 1 (bottom) is exact in the sense that u n j equals the average of the exact solution on C j , that is,…”

“…We rely on the discontinuous reconstruction technique proposed recently in Lagoutière [21,22] which has been found to be particularly efficient for computing classical solutions of (1) with moderate numerical diffusion.…”

Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for Transport

“…Details and proofs can be found in (Lagoutière, 2005) with (Lagoutière, 2006). We consider a mesh (on R) with a constant space step Δx > 0 whose cells are the intervals T j = ((j − 1/2)Δx, (j + 1/2)Δx) for j ∈ Z.…”

“…It is based on the geometrical approach followed in (Lagoutière, 2005;Lagoutière, 2006), which provided a new interpretation of the limited downwind algorithm in terms of reconstruction schemes.…”

“…The proposed reconstruction, which is genuinely multidimensional, allows recovering sharp profiles in both the direction of the transport velocity and the transverse direction. It constitutes an extension of the one-dimensional reconstructions analyzed in (Lagoutière, 2005;Lagoutière, 2006). …”