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Richard De la cruz

Publications and source records attributed to Richard De la cruz.

6 recordsLinked to original sources
arXiv2026

Wave interactions and stability of Riemann solutions for a nonautonomous Chromatography-type system of Langmuir isotherm

We investigate the wave interactions and stability of Riemann solutions for a nonautonomous chromatography-type system of Langmuir isotherm with time-dependent damping and flux. The system models two-component chromatographic separation with a time-dependent saturation capacity $n(t)$, leading to a nonautonomous hyperbolic system of balance laws. We consider a perturbed Riemann problem with piecewise constant initial data having two jump discontinuities at $x = \pmε$, and construct the global weak solution by analyzing all possible wave interactions, both classical (shock waves, rarefaction waves, contact discontinuities) and nonclassical (delta shock waves). We prove that as $ε\to 0$, the solution of the perturbed Riemann problem converges to the solution of the corresponding Riemann problem in the space of Radon measures, establishing the stability of Riemann solutions under small perturbations of the initial data. To the best of our knowledge, this is the first instance of wave interaction and stability analysis for a nonautonomous chromatography-type system with time-dependent coefficients. Numerical experiments using a Lax-Friedrichs type scheme illustrate the wave interaction structure, the profiles at selected times, and the asymptotic convergence as $ε\to 0$.

math.AP↗
arXiv2024

A Semidiscrete Lagrangian-Eulerian scheme for the LWR traffic model with discontinuous flux

In this work, we present a semi-discrete scheme to approximate solutions to the scalar LWR traffic model with spatially discontinuous flux, described by the equation $u_t + (k(x)u(1-u))_x = 0$. This approach is based on the Lagrangian-Eulerian method proposed by E. Abreu, J. Francois, W. Lambert, and J. Perez [J. Comp. Appl. Math. 406 (2022) 114011] for scalar conservation laws. We derive a non-uniform bound on the growth rate of the total variation for approximate solutions. Since the total variation can explode only at $x=0$, we can provide a convergence proof for our scheme in $BV_{loc}(\mathbb{R}\setminus \lbrace 0 \rbrace)$ by using Helly's compactness theorem.

math.NA↗
arXiv2024

Riemann problem for a nonsymmetric Keyfitz-Kranzer and pressureless gas systems with a time-dependent Coulomb-like friction term

In this paper, we study the Riemann solutions for two systems: the nonsymmetric Keyfitz-Kranzer system and the pressureless system, both of which have a time-dependent Coulomb-like friction term. Our analysis identified two types of Riemann solutions: contact discontinuities and delta-shock solutions. We obtain generalized Rankine-Hugoniot conditions, which is the support for constructing the delta-shock solution for the nonsymmetric Keyfitz-Kranzer system with a time-dependent Coulomb-like friction term. Furthermore, we demonstrate that as the pressure tends to zero, the Riemann solutions of the nonsymmetric Keyfitz-Kranzer system converge to those of the pressureless system, with both systems incorporating a time-dependent Coulomb-like friction term.

math.AP↗
arXiv2024

On a class of nonautonomous quasilinear systems with general time-gradually-degenerate damping

In this paper, we study two systems with a time-variable coefficient and general time-gradually-degenerate damping. More explicitly, we construct the Riemann solutions to the time-variable coefficient Zeldovich approximation and time-variable coefficient pressureless gas systems both with general time-gradually-degenerate damping. Applying the method of similar variables and nonlinear viscosity, we obtain classical Riemann solutions and delta shock wave solutions.

math.AP↗
arXiv2015

Delta shock wave for a $3 \times 3$ hyperbolic system of conservation laws

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characteristic field is linearly degenerate. We show the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.

math.AP↗