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Eduardo Abreu

Publications and source records attributed to Eduardo Abreu.

12 recordsLinked to original sources
arXiv2026

Complex Dynamics of Wave-Character Transitions in Radially Symmetric Isentropic Euler Flows: Theory and Numerics

We investigate the qualitative dynamics of smooth solutions to the radially symmetric isentropic compressible Euler equations, focusing specifically on the evolution of rarefactive and compressive wave characters across three distinct configurations: the outward supersonic, subsonic, and inward supersonic regimes. For each case, we establish structural restrictions on wave-character transitions and identify invariant sign domains for gradient variables under specific initial data conditions. While our findings refine existing invariance properties in the outward supersonic regime, they reveal novel asymmetric transition mechanisms in the subsonic and inward regimes that are absent in purely supersonic expanding cases. Consequently, we derive sufficient conditions for finite-time singularity formation. To complement the analytical results where closed-form solutions are unavailable, we provide numerical experiments using a Semi-Discrete Lagrangian-Eulerian (SDLE) formulation. These simulations reproduce the predicted wave-character dynamics and offer qualitative evidence that supports our theoretical findings, providing a unified description of wave transitions in radially symmetric isentropic gas dynamics.

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↗
arXiv2022

A geometrically intrinsic Lagrangian-Eulerian scheme for 2D Shallow Water Equations with variable topography and discontinuous data

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme maintains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometrically intrinsic shallow water equation on variable topographies. We provide a representative set of numerical examples to illustrate the accuracy and robustness of the proposed Lagrangian-Eulerian formulation for two-dimensional surfaces with general curvatures and discontinuous initial conditions.

math.NA↗
arXiv2022

Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions. We present experiments to a study based on the use of the Wasserstein distance to show the effectiveness of the no-flow curves approach in the cases of shock interaction with an entropy wave related to the inviscid Burgers' model problem and to a 2x2 nonlocal traffic flow symmetric system of type Keyfitz--Kranzer.

math.NA↗
arXiv2021

A study on a feedforward neural network to solve partial differential equations in hyperbolic-transport problems

In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account the equation and initial conditions of the model. We apply it to the Riemann problems over the inviscid nonlinear Burger's equation, whose solutions might develop discontinuity (shock wave) and rarefaction, as well as to the classical one-dimensional Buckley-Leverett two-phase problem. The Buckley-Leverett case is slightly more complex and interesting because it has a non-convex flux function with one inflection point. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in such challenging tasks, providing numerical approximation of entropy solutions with very good precision and consistent to classical as well as to recently novel numerical methods in these particular scenarios.

math.NA↗
arXiv2021

An application of a pseudo-parabolic modeling to texture image recognition

In this work, we present a novel methodology for texture image recognition using a partial differential equation modeling. More specifically, we employ the pseudo-parabolic Buckley-Leverett equation to provide a dynamics to the digital image representation and collect local descriptors from those images evolving in time. For the local descriptors we employ the magnitude and signal binary patterns and a simple histogram of these features was capable of achieving promising results in a classification task. We compare the accuracy over well established benchmark texture databases and the results demonstrate competitiveness, even with the most modern deep learning approaches. The achieved results open space for future investigation on this type of modeling for image analysis, especially when there is no large amount of data for training deep learning models and therefore model-based approaches arise as suitable alternatives.

cs.CV↗
arXiv2021

Texture image classification based on a pseudo-parabolic diffusion model

This work proposes a novel method based on a pseudo-parabolic diffusion process to be employed for texture recognition. The proposed operator is applied over a range of time scales giving rise to a family of images transformed by nonlinear filters. Therefore each of those images are encoded by a local descriptor (we use local binary patterns for that purpose) and they are summarized by a simple histogram, yielding in this way the image feature vector. The proposed approach is tested on the classification of well established benchmark texture databases and on a practical task of plant species recognition. In both cases, it is compared with several state-of-the-art methodologies employed for texture recognition. Our proposal outperforms those methods in terms of classification accuracy, confirming its competitiveness. The good performance can be justified to a large extent by the ability of the pseudo-parabolic operator to smooth possibly noisy details inside homogeneous regions of the image at the same time that it preserves discontinuities that convey critical information for the object description. Such results also confirm that model-based approaches like the proposed one can still be competitive with the omnipresent learning-based approaches, especially when the user does not have access to a powerful computational structure and a large amount of labeled data for training.

cs.CV↗
arXiv2020

On the conservation properties in multiple scale coupling and simulation for Darcy flow with hyperbolic-transport in complex flows

We present and discuss a novel approach to deal with conservation properties for the simulation of nonlinear complex porous media flows in the presence of: 1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model, and 2) multiscale wave structures resulting from shock waves and rarefaction interactions from the nonlinear hyperbolic-transport model. For the pressure-velocity Darcy flow problem, we revisit a recent high-order and volumetric residual-based Lagrange multipliers saddle point problem to impose local mass conservation on convex polygons. We clarify and improve conservation properties on applications.For the hyperbolic-transport problem we introduce a newlocally conservative Lagrangian-Eulerian finite volume method. For the purpose of this work, we recast our method within the Crandall and Majda treatment of the stability and convergence properties of conservation-form, monotone difference, in which the scheme converges to the physical weak solution satisfying the entropy condition. This multiscale coupling approach was applied to several nontrivial examples to show that we are computing qualitatively correct reference solutions. We combine these procedures for the simulation of the fundamental two-phase flow problem with high-contrast multiscale porous medium, but recalling state-of-the-art paradigms on the of notion of solution in related multiscale applications. This is a first step to deal with out-of-reach multiscale systems with traditional techniques. We provide robust numerical examples for verifying the theory and illustrating the capabilities of the approach being presented.

math.NA↗
arXiv2020

On the use of spectral discretizations with time strong stability preserving properties to Dirichlet pseudo-parabolic problems

This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation methods based on Legendre and Chebyshev polynomials. The time integration is carried out suitably with robust schemes attending to qualitative features such as stiffness and preservation of strong stability to simulate nonregular problems more correctly. The corresponding semidiscrete and fully discrete schemes are described and the performance of the methods is analyzed computationally.

math.NA↗
arXiv2019

A convergence analysis of Generalized Multiscale Finite Element Methods

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of [J. Comput. Phys. 230 (2011), 937-955]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis in view of the good scientific computing practice.

math.NA↗
arXiv2017

On high-order conservative finite element methods

A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred.

math.NA↗