Representation of Weak Limits and Definition of Nonconservative Products

Abstract

The goal of this article is to show that the notion of generalized graphs is able to represent the limit points of the sequence {g(un) dun} in the weak-⋆ topology of measures when {un} is a sequence of continuous functions of uniformly bounded variation. The representation theorem induces a natural definition for the nonconservative product g(u) du in a BV context. Several existing definitions of nonconservative products are then compared, and the theory is applied to provide a notion of solutions and an existence theory to the Riemann problem for quasi-linear, strictly hyperbolic systems.

Localization effects and measure source terms in numerical schemes for balance laws

“…The general results of [12,32,36] guarantee uniqueness for these Riemann problems, but the next lemma establishes uniqueness for the Cauchy problem (49) under some restriction on the right-hand side. …”

“…Indeed, w ε and z ε suffer discontinuities at least on the lines t = n∆t, n ∈ N * , and their product with a Dirac mass has to be defined in a careful way. Anyway, we can give a precise meaning to this kind of terms in the framework proposed in [32,36] by investigating the limit of the sequence G(w ε , z ε )∂ t b ε in the weak-topology of measures on R × R + . The next result lies at the heart of the matter.…”

“…Now we want to shed some light on the ambiguous term emanating from the limit ε → 0 of (34). Once again, we will use nonconservative products as defined in [32,36], that is, as the weak-limits of the compact sequences g(u ε )∂ x a ε .…”

“…Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in [5,9,12,24,32,36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17,19,20,22,23]; see also [8,21,40,39] for closely related works.…”

Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients

“…([1], [2]), Fornet ([8], [9]), Gallouët([10]), LeFloch and al. ([6], [14], [15], [13]). The common idea is that another notion of solution has to be introduced to deal with linear hyperbolic Cauchy problems with discontinuous coefficients.…”