The Spaces $ Bv$ and Quasilinear Equations

Abstract

This article consists of two parts. The first ( § §2-5) is devoted to some questions of analysis in the space of functions whose generalized derivatives are measures (the spaces BV). The results obtained here are applied in the second part ( § §6-7) to the study of quasilinear equations. In §6 we prove a theorem on the existence in the large and the uniqueness of the solution of the Cauchy problem for a first-order quasilinear equation. In §7 we consider quasilinear operators in nondivergence form acting on discontinuous functions in BV.

On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

“…The simplest choice is given by the family of segments: (2.9) that corresponds to the definition of nonconservative product proposed by Volpert [38]. In general, in practical applications, its selection has to be based on the physical background (see [22,28] …”

High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems

“…In this case, the choice of an appropriate family of paths to define the weak solutions of the problem based on a regularization of the problem is a difficult task. With this numerical scheme, which is based on the family of segments, the speeds of the shocks related to the genuinely nonlinear fields given by the scheme are expected to fit to Volpert's definition of nonconservative products (see [22]) which is equivalent to the definition corresponding to the family of segments, i.e. they are expected to fit to the Rankine-Hugoniot condition (4) corresponding to the family of segments.…”

“…that corresponds to the definition of nonconservative products proposed by Volpert (see [22]). In practical applications, it has to be based on the physical background of the problem.…”

Efficient simulation of one-dimensional two-phase flow with a high-order h-adaptive space-time Discontinuous Galerkin method

“…Since the system of PDEs is non-conservative, at least for the first model described in Section 2 , standard Riemann solvers cannot be applied. Vol'pert [21] studied non-conservative systems and interpreted the non-conservative product as a product of a function with a measure. Dal Maso et al [22] generalised this interpretation of the non-conservative product, known as the DLM-measure .…”