arXiv subjects

Bradley J. Plohr

Publications and source records attributed to Bradley J. Plohr.

3 recordsLinked to original sources

arXiv2026

When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants?

A solution of a Riemann problem for a strictly hyperbolic system of conservation laws is traditionally expected to consist of rarefaction waves, jump discontinuities, and constant states. In this paper, we investigate whether a Riemann solution has this structure when the solution is only assumed to be measurable and essentially bounded. To discriminate continuous and discontinuous features in an $L^\infty$ solution, we introduce one-sided accumulation sets based on local essential images. Supposing that throughout a bounded open interval a solution is continuous in the essential image (ess-im) sense, we prove that it is a rarefaction wave if it is resonant (the characteristic speed equals $x/t$), and otherwise it is constant. Although an ess-im discontinuity might not be a jump discontinuity, we show that all ess-im accumulation states lie on a common Hugoniot locus and have the same speed. Anomalies are possible if there are limit points of ess-im discontinuities, but if the set of ess-im discontinuities is finite, then an $L^\infty$ Riemann solution has bounded variation and is composed of finitely many rarefaction waves, jump discontinuities, and constant states.

math.AP↗

arXiv2024

Solving Riemann Problems with a Topological Tool (Extended version)

In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.

math.AP↗

arXiv2024

Vanishing adsorption limit of Riemann problem solutions for the polymer model

We examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm-Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz-Kranzer, Isaacson-Temple, and de Souza-Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

math.AP↗