The following work have been accepted for publication in "Transport in Porous Media" and is now available online:

An Enhanced Reduced Flow Model for Paleokarst Reservoirs Incorporating Multi-stage Collapse Breccia Pipes

https://link.springer.com/article/10.1007/s11242-025-02188-y

E. Abreu, J. A. Barbosa, I. Landim, M. A. Murad, P. Pereira

Abstract:

We develop an innovative mixed-dimensional 3D/1D flow model in carbonate rocks containing multiple karst cave conduits with underlying heterogeneity in the petrophysical properties stemming from different geological stages of cave-pipe collapse systems. Such geological structures manifest in distinct heterogeneity patterns inherent to the successive stages of burial, mechanical failure, and collapse, resulting in discrete collapsed passages in the conduit network. In addition, breakdown products appear within the cave system associated with chaotic breccia, suprastrata deformation, and vertical tube-like geo-bodies, herein referred to as breccia pipes, containing faults and fractures around the vertical pipe. The input parameters of the mixed-dimensional flow model show the ability to incorporate the complex multiple heterogeneities associated with the geological objects at different stages of collapse. After populating the geo-bodies with proper petrophysical properties, the mixed-dimensional flow equations are discretized by a locally conservative extended version of the mixed-hybrid finite element method, which incorporates the new nonlinear discrete transmission jump conditions between elements adjacent to the breccias within the conduits. Computational simulations are performed for particular configurations of heterogeneous karst conduit systems with distinct geological time scales, illustrating the influence of the karst and solution breccia-pipe deposits upon the flow regimes, streamline patterns, and well productivity in real-case scenarios of hypogenic cave networks.

 

We are pleased to announce that Eduardo Cardoso de Abreu, professor at the Department of Applied Mathematics of the State University of Campinas UNICAMP (Brazil), will give the SFB 1313 "Pretty Porous Science Lecture" #66. His talk will be on "A novel forward Lagrangian-Eulerian method for computing hyperbolic PDEs and applications".

Date: Tuesday, 17 June 2025

Time: 4 pm Stuttgart (11 am Brazil)

Speaker: Prof. Eduardo Cardoso de Abreu, State University of Campinas UNICAMP (Brazil)

Title: "A novel forward Lagrangian-Eulerian method for computing hyperbolic PDEs and applications"

Venue: Multi Media Lab (MML), U1.003, Pfaffenwaldring 61, 70569 Stuttgart, Campus Vaihingen. If you are interested in participating in the lecture online, please contact samaneh.vahiddastjerdi@mechbau.uni-stuttgart.de

Abstract:

Modeling, mathematical and numerical analysis challenges for the study of hyperbolic PDEs is in the realm of basic and applied sciences related to fluid mechanics, modeling of vehicular traffic flows, and fluid dynamics  in porous media flows, just to name a few specific problems. We design a new class of fully-discrete and semi-discrete Lagrangian-Eulerian schemes to approximate nonlinear multidimensional initial value problems for scalar models and multidimensional systems of hyperbolic conservation laws. The approach is also applied to nonlinear balance laws. The method is based on the concept of multidimensional no-flow curves/surfaces/manifolds. Roughly speaking, one reduces the hyperbolic PDE into a family of ODEs along the forward untangled space-time no-flow Lagrangian trajectories. Due to the no-flow framework, there is no need to compute the eigenvalues (exact or approximate values), and in fact there is no need to construct the Jacobian matrix of the hyperbolic flux functions, and thus giving rise to an effective weak CFL-stability condition in the computing practice. We were able to provide a convergence proof towards the entropy solution to the scalar problem. In the case of multidimensional hyperbolic systems of conservation laws, we show that the Lagrangian-Eulerian scheme also satisfies the weak version of the positivity principle proposed by P. Lax and X.-D. Liu. We also found a connection between some of the results of A. Bressan, in the context of local existence and continuous dependence for discontinuous ODEs and the no-flow curves (now viewed as a forward vector field with locally bounded variation). We present numerical computations for nontrivial (local and nonlocal) hyperbolic problems, as such compressible Euler flows with positivity of the density, the Orszag-Tang problem, which is well-known to satisfy the notable involution-constrained partial differential equation div B = 0, a nonstrictly hyperbolic three-phase flow system in porous media with a resonance point, and the classical 3 by 3 shallow-water system (with and without discontinuous bottom topography). We will also briefly present some numerical results in the context of high-performance parallel computing via a MPI environment.

  

NumHyp25: Numerical Methods for Hyperbolic Problems, Darmstadt, Germany, 9th to 13th June 2025

https://numhyp25.sciencesconf.org/

NumHyp is a conference aiming to promote exchange between mathematicians from different countries in Europe on issues related to recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations. These equations appear in a large number of models in science and engineering. Some of the best known examples are the compressible Euler and Navier-Stokes equations, shallow water equations, magnetohydrodynamic equations, multiphase fluid models, hyperbolic formulations of continuum mechanics, and even general relativity. Examples of application areas include aerodynamics, oceanography, plasma physics, solid mechanics or computational astrophysics.

Main topics of the NumHyp25 include:

High-order methods

Uncertainty Quantification

Measure-valued/statistical solutions

Stochastic models

Multiscale models, Kinetic equations

Asymptotic preserving methods

Well-balanced methods

Applications: geophysics, atmospheric flows, astrophysics, etc.

EDUARDO ABREU presentation starts monday, 09th June, at 12:00am:

Fully-discrete and semi-discrete Lagrangian-Eulerian formulations for hyperbolic systems of conservation laws in three-space dimensions on structured cubical and tetrahedral meshes

ABSTRACT: https://numhyp25.sciencesconf.org/data/program/1.pdf

 

International Congress of Mathematicians (ICM) 2026 , Philadelphia, USA, 23th to 30th July 2026

https://www.mathunion.org/icm/icm-2026

At its meeting in Helsinki, Finland, in July 2022, the 19th IMU General Assembly voted to accept the bid from the United States of America to host ICM 2026 and the 20th IMU General Assembly, with Philadelphia as venue for the ICM and New York City for the General Assembly.

ICM 2026 will be hosted at the Pennsylvania Convention Center in Philadelphia over 23–30 July 2026.

The 20th IMU General Assembly will convene at the Marriott Marquis Times Square over 20–21 July 2026.

The official website of the Congress is www.icm2026.org. The poster for ICM 2026 can be found here.

Further information regarding ICM 2026 will be made available in due course.

ENUMATH: European Conference on Numerical Mathematics and Advanced Applications, Heidelberg, Germany, 1st to 5th September 2025

https://enumath2025.eu/

The European Conference on Numerical Mathematics and Advanced Applications (ENUMATH) conferences are a forum for presenting and discussing novel and fundamental advances in numerical mathematics and challenging scientific and industrial applications on the highest level of international expertise.

Conference Themes:

-Advances in Discretisation Schemes

-Multiscale and Multiphysics Problems

-Hardware-Aware Scientific Computing

-Inverse Problems

-Uncertainty Quantification 

-Data-Driven Modelling and Simulation

-Scientific Machine Learning

-Reduced Order Models and Surrogates

-Randomised Numerical Algorithms

-Numerical Optimisation and Optimal Control

EDUARDO ABREU, ELENA BACHINI, JONH PEREZ AND MARIO PUTTI will be presenting the minisymposia: MS67 - Advances in Lagrangian-Eulerian Schemes for Hyperbolic Systems of Conservation Laws.

 

NumHyp25: Numerical Methods for Hyperbolic Problems, Darmstadt, Germany, 9th to 13th June 2025

https://numhyp25.sciencesconf.org/

NumHyp is a conference aiming to promote exchange between mathematicians from different countries in Europe on issues related to recent developments and new directions in the field of numerical methods for hyperbolic partial differential equations. These equations appear in a large number of models in science and engineering. Some of the best known examples are the compressible Euler and Navier-Stokes equations, shallow water equations, magnetohydrodynamic equations, multiphase fluid models, hyperbolic formulations of continuum mechanics, and even general relativity. Examples of application areas include aerodynamics, oceanography, plasma physics, solid mechanics or computational astrophysics.

Main topics of the NumHyp25 include:

High-order methods

Uncertainty Quantification

Measure-valued/statistical solutions

Stochastic models

Multiscale models, Kinetic equations

Asymptotic preserving methods

Well-balanced methods

Applications: geophysics, atmospheric flows, astrophysics, etc.

EDUARDO ABREU presentation starts monday, 09th June, at 12:00am:

Fully-discrete and semi-discrete Lagrangian-Eulerian formulations for hyperbolic systems of conservation laws in three-space dimensions on structured cubical and tetrahedral meshes

ABSTRACT: https://numhyp25.sciencesconf.org/data/program/1.pdf

The following work is now available online in the April 2025 volume "Journal of Computational and Applied Mathematics" https://doi.org/10.1016/j.cam.2024.116325:

Semi-discrete Lagrangian-Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions

Eduardo Abreu, Richard A. De la Cruz Guerrero, Juan Carlos Juajibioy Otero and Wanderson José Lambert.

ABSTRACT:
In this work, we have expanded upon the (local) semi-discrete Lagrangian–Eulerian method initially introduced in Abreu et al. (2022) to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux, referred to as the nonlinear nonlocal model:

For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in Abreu et al. (2016), with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian–Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space 𝐿¹(Rᵈ) ∩𝐿∞(Rᵈ). 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the 𝐿¹-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution of Eq. (1). Finally, we present a section of numerical examples to illustrate our results. In particular, we have examined examples discussed in Aggarwal et al. (2015) and Keimer et al. (2018). In addition, we also provide numerical results for a nonlocal impact of the form 𝜔𝜂 ∗ 𝜌, where 𝜂=0.1 for class of the two-dimensional nonlinear nonlocal inviscid Burgers’ equations. 

 

SIAM Conference on Mathematical & Computational Issues in the Geosciences , Baton Rouge, USA, 14th to 17th October 2025

https://www.siam.org/conferences-events/siam-conferences/gs25/

The study of geophysical systems, whether from a scientific or technological perspective, calls for sophisticated mathematical modeling, efficient computational methods, and pervasive integration with data. This conference aims to stimulate the exchange of ideas among geoscientific modelers, applied mathematicians, engineers, and other scientists, having special interests in the range of geophysical domains from the deep subsurface to the atmosphere.

Included Themes:

-Mathematical and computational research in geoscience at all scales:

-Applications to porous media systems, geophysics, reservoir engineering, geologic sequestration, coastal engineering, water resources, and ecology

-Data assimilation and machine learning

-Mathematical models and numerical analysis

-Solvers and scientific computing

-Uncertainty quantification

The following conference paper is now avaiable online:

Blowing Up and Dissipation for a Couple of One-dimensional Non-local Conservation Laws

E. Abreu, J. C. Valencia-Guevara, M. Huacasi-Machaca & J. Pérez.

LINK: https://link.springer.com/chapter/10.1007/978-3-031-77050-0_20

Conference paper: First Online: 30 January 2025,  pp 263–278.

Included in the following conference series: ISAAC Congress (International Society for Analysis, its Applications and Computation).

The thematic session, "Partial Differential Equations - Methods, Computing and Applications", organized by Eduardo Abreu (UNICAMP, Brazil), Patricia Saavedra Barrera (Universidad Autónoma Metropolitana, Mexico) has been accepted for the First Joint Meeting Brazil-Mexico in Mathematics 2025.

SITE: https://sbm.org.br/jointmeeting-mexico/.

DATE:  September 8th to 12th 2025

WHERE:  Fortaleza, CE, Brazil

The Brazilian Mathematical Society (SBM) and the Brazilian Society for Computational and Applied Mathematics (SBMAC) are honored to invite the mathematical community to take part in the first Brazil-Mexico Joint Mathematical Meeting, which will be held in Fortaleza, Brazil, from September 8 to 12, 2025. This event is a partnership with the Mexican Mathematical Society (SMM) https://sbm.org.br/jointmeeting-mexico/welcome/

International Congress of Mathematicians (ICM) 2026 , Philadelphia, USA, 23th to 30th July 2026

https://www.mathunion.org/icm/icm-2026

ICM 2026 will be hosted at the Pennsylvania Convention Center in Philadelphia over 23–30 July 2026.

The 20th IMU General Assembly will convene at the Marriott Marquis Times Square over 20–21 July 2026.

International Mathematical Union (IMU) Home ==> https://www.mathunion.org/

Mathematical Congress of the Americas 2025: July 21, 2025 – July 25, 2025

The goal of the Mathematical Congress of the Americas (MCA) is to internationally highlight the excellence of mathematical achievements in the Americas and foster collaborations among researchers, students, institutions and mathematical societies in the Americas.

LINK: https://www.mca2025.org/event/9e9666dd-2643-423b-b343-91f10f36e686/summary

Special Sessions: https://www.mca2025.org/event/9e9666dd-2643-423b-b343-91f10f36e686/websitePage:f3552742-6e9a-44e3-a319-6b6056ba9900

In the Mathematical Congress of the Americas (MCA), take a look at

Session 38: Conservation Laws: Mathematical and Numerical Analysis with Applications

Organizers:

Eduardo Abreu (Universidade Estadual de Campinas, Brazil, contact organizer)
Fabio Ancona (University of Padua, Italy)
Maria Teresa Chiri (Queen’s University, Canada)
Xiaoqian Gong (Amherst College, USA)
Michael Herty (RWTH Aachen University, Germany)

Brief Summary: Hyperbolic conservation laws have been subject to extensive analytical and numerical studies over the last decades. It is widely known that their solutions can exhibit very complex behavior including the simultaneous presence of smooth waves, wave breaking, and shock waves. These equations describe the conservation of some basic physical quantities of a system, and they arise in all branches of science and engineering: from fluid dynamics to vehicular traffic modeling. The scope of this Special Session is to bring together researchers with interests in the theoretical, applied, and computational aspects of hyperbolic partial differential equations with real-life applications and discuss the state of the art of the field.

In the perspective of accomplishing the goals of the 2025 Mathematical Congress of the Americas (MCA2025), the invited speakers are of different ages, nationalities, and different scientific career stages and they are selected among the leaders in the field. This aspect makes the Special Session suitable for training young researchers and fostering interactions between several mathematical communities across the Americas (South, Central and North).

The following work have been accepted for publication on "CALCOLO" (https://link.springer.com/article/10.1007/s10092-024-00624-x):

A numerical scheme for doubly nonlocal conservation laws

E. Abreu, J. C. Valencia-Guevara, M. Huacasi-Machaca & J. Pérez.

Published: 22 November 2024; Volume 61, article number 72, (2024).

Keywords: Fractional conservation laws. Doubly nonlinear nonlocal flux. Riesz potential. Hilbert transform. Numerical algorithm for the Riesz fractional Laplacian. Nonlocal Lagrangian–Eulerian scheme. Nonlocal no-flow curves.

Mathematics Subject Classification: 47G40. 47B34. 65Y20. 35R11. 35B44.

Highlights:
In this work, we consider the nonlinear dynamics and computational aspects for nonnegative solutions of one-dimensional doubly nonlocal fractional conservation laws

∂tu + ∂x [uA^α−1(κ(x)Hu)] = 0 and ∂tu − ∂x [uA^α−1(κ(x)Hu)] = 0,

where A^α−1 denotes the fractional Riesz transform, H denotes the Hilbert transform, and κ(x) denotes the spatial variability of the permeability coefficient in a porous medium. We construct an unconventional Lagrangian–Eulerian scheme, based on the concept of no-flow curves, to handle the doubly nonlocal term, under a weak CFL stability condition, which avoids the computation of the derivative of the nonlocal flux function. Primarily, we develop a feasible computational method and derive error estimates of the approximations of the Riesz potential operator A^α−1. Secondly, we undertake a formal numerical-analytical study of initial value problems associated with such doubly nonlocal models to add insights into the role of nonlinearity and coefficient κ(x) in the composition between the Hilbert transform and the fractional Riesz potential. Numerical experiments are presented to show the performance of the approach.

 

25 anos do Bacharelado em Matemática Computacional da UFMG

Semana da MatComp: 18, 19 e 21 de novembro de 2024

Dia Comemorativo: 22 de novembro de 2024

HOMEPAGE:  https://www.matcomp25anos.dcc.ufmg.br/

Palestra Convidada: "Eduardo Abreu: Análise numérica para modelos diferenciais: sinergia natural entre contínuo e discreto na matemática e aplicações"

LINK:  https://www.matcomp25anos.dcc.ufmg.br/#palestras

The following work have been accepted for publication on "Numerical Methods for Partial Differential Equations" and it will available online soon (MNPDE, https://onlinelibrary.wiley.com/journal/10982426):

An enhanced Lagrangian-Eulerian method for a class of balance Laws: numerical analysis via a weak asymptotic method with applications

Eduardo Abreu, Eduardo Pandini and Wanderson José Lambert.

Highlights:
In this work, we designed and implemented an enhanced Lagrangian-Eulerian numerical method for solving a wide range of nonlinear balance laws, including systems of hyperbolic equations with source terms. We developed both fully discrete and semi-discrete formulations, and extended the concept of No-Flow curves to this general class of nonlinear balance laws. We conducted a numerical convergence study using weak asymptotic analysis, which involved investigating the existence, uniqueness, and regularity of entropy-weak solutions computed with our scheme. The proposed method is Riemann-solver-free. To evaluate the shock capturing capabilities of the enhanced Lagrangian-Eulerian  numerical scheme, we carried out numerical experiments that demonstrate its ability to accurately resolve the key features of balance law models and hyperbolic problems. A representative set of numerical examples is provided to illustrate the accuracy and robustness of the proposed method.

ISAAC-ICMAM Conference of Analysis in Developing Countries 2024.

https://www.matua.edu.co/isaac-icmam-conference-of-analysis-in-developing-countries-2024/

The ISAAC-ICMAM (Virtual) Conference of Analysis in Developing Countries represents a significant collaboration between ISAAC and ICMAM Latin America. This jointly organized conference aims to share mathematical research in Latin America and the Caribbean, enhance its visibility, and foster collaboration among mathematicians from the region and worldwide.

During the conference, the following speak will be presented by Eduardo Abreu (Universidade Estadual de Campinas, UNICAMP, Brazil):

Title: A forward-tracking Lagrangian-Eulerian method for multidimensional systems of conservation laws

Abstract: We will discuss a forward Lagrangian-Eulerian approach to undertake a numerical-analytical study of inherent properties of multidimensional nonlinear hyperbolic conservation laws [(2024)  https://doi.org/10.1016/j.cam.2023.115465]. It is widely known that their solutions can exhibit very complex behavior including the simultaneous presence of smooth waves, wave breaking, and shock waves. The novel forward tracking Lagragian-Eulerian formulation is based on the improved concept of no-flow curves. In the context of multidimensional hyperbolic systems of conservation laws, the resulting Lagrangian-Eulerian method satisfies a weak positivity principle in view of results of P. Lax and X.-D. Liu [Computational Fluid Dynamics Journal, 5(2) (1996) 133-156 and [Journal of Computational Physics, 187 (2003) 428-440]. We also found in [(2023) https://doi.org/10.1007/s10884-023-10283-1] a connection between the notion of no-flow curves, viewed as a vector field with locally bounded variation, and the results of A. Bressan in the context of existence and continuous dependence for discontinuous O.D.E.’s [Proc. Amer. Math. Soc. 104 (1988), 772-778]. We have tested the approach for well-known non-trivial muitlt-D systems and complex problems in fluid dynamics [(2023) https://doi.org/10.1016/j.amc.2022.127776,  (2023) https://link.springer.com/article/10.1007/s10915-021-01712-8 , (2021) https://doi.org/10.1007/s10915-020-01392-w]: 4 by 4 compressible Euler equations (Double Mach Reflection problem and Mach 3 wind tunnel flow, the 3 by 3 shallow-water system with and without bottom topography, and the 8 by 8 Orszag-Tang vortex system in magnetohydrodynamics and a nonclassical 2 by 2 three-phase flow system of non strictly hyperbolic conservation laws with a resonance/umbilic point.

ISAAC-ICMAM Conference of Analysis in Developing Countries 2024:

  

Matemáticas & Memoria: Un seminario de Análisis y Ecuaciones diferenciales para Latinoamérica.

Matemáticas y Memoria: Un seminario de Análisis y Ecuaciones diferenciales para Latinoamérica. – MATUA

El seminario «Matemáticas y memoria: un seminario de Análisis y Ecuaciones diferenciales para Latinoamérica» tiene como objetivo principal promover el intercambio de conocimientos y experiencias en el campo del análisis y las ecuaciones diferenciales, destacando la importancia de la memoria histórica en el desarrollo matemático de Latinoamérica. A través de la participación de invitados nacionales e internacionales, la visibilización de jóvenes talentos matemáticos y la invitación a reconocidos investigadores establecidos, se busca fomentar el diálogo académico y fortalecer la comunidad matemática regional.

Presentacíon de octubre: No-Flow Lagrangian-Eulerian Curves for Hyperbolic Conservation Laws. Eduardo Abreu (Universidade Estadual de Campinas, Brasil). LINK:  https://youtu.be/HQMibFXNnUg

Se puede acceder a las demás presentaciones en Matemáticas y Memoria: Un seminario de Análisis y Ecuaciones diferenciales para Latinoamérica. – MATUA.

Matemáticas y memoria: un seminario de Análisis y Ecuaciones diferenciales para Latinoamérica:

The following work have been accepted for publication on "Journal of Computational and Applied Mathematics" https://doi.org/10.1016/j.cam.2024.116325:

Semi-discrete Lagrangian-Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions

Eduardo Abreu, Richard A. De la Cruz Guerrero, Juan Carlos Juajibioy Otero and Wanderson José Lambert.

Highlights:
In this work, we have expanded upon the (local) semi-discrete Lagrangian-Eulerian method initially introduced in [E. Abreu, J. Francois, W. Lambert and J. Perez (2022), https://doi.org/10.1016/j.cam.2021.114011] to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux. For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in [Eduardo Abreu, Mathilde Colombeau and Evgeny Yu Panov (2016), https://doi.org/10.1016/j.jmaa.2016.06.047], with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian-Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space L1(Rd) ∩ L∞(Rd) and 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the L1-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution. Finally, we present a section of numerical examples in 1D and also in 2D for two-dimensional nonlocal Burgers equations to illustrate our results.

 

Mathematical Congress of the Americas 2025: July 21, 2025 – July 25, 2025

The goal of the Mathematical Congress of the Americas (MCA) is to internationally highlight the excellence of mathematical achievements in the Americas and foster collaborations among researchers, students, institutions and mathematical societies in the Americas.

LINK: https://www.mca2025.org/event/9e9666dd-2643-423b-b343-91f10f36e686/summary

Special Sessions: https://www.mca2025.org/event/9e9666dd-2643-423b-b343-91f10f36e686/websitePage:f3552742-6e9a-44e3-a319-6b6056ba9900

In the Mathematical Congress of the Americas (MCA), take a look at

Session 38: Conservation Laws: Mathematical and Numerical Analysis with Applications

Organizers:

Eduardo Abreu (Universidade Estadual de Campinas, Brazil, contact organizer)
Fabio Ancona (University of Padua, Italy)
Maria Teresa Chiri (Queen’s University, Canada)
Xiaoqian Gong (Amherst College, USA)
Michael Herty (RWTH Aachen University, Germany)

Brief Summary: Hyperbolic conservation laws have been subject to extensive analytical and numerical studies over the last decades. It is widely known that their solutions can exhibit very complex behavior including the simultaneous presence of smooth waves, wave breaking, and shock waves. These equations describe the conservation of some basic physical quantities of a system, and they arise in all branches of science and engineering: from fluid dynamics to vehicular traffic modeling. The scope of this Special Session is to bring together researchers with interests in the theoretical, applied, and computational aspects of hyperbolic partial differential equations with real-life applications and discuss the state of the art of the field.

In the perspective of accomplishing the goals of the 2025 Mathematical Congress of the Americas (MCA2025), the invited speakers are of different ages, nationalities, and different scientific career stages and they are selected among the leaders in the field. This aspect makes the Special Session suitable for training young researchers and fostering interactions between several mathematical communities across the Americas (South, Central and North).

The following work have been accepted for publication on "Computers and Mathematics with Applications" https://doi.org/10.1016/j.camwa.2024.04.015

Mathematical properties and numerical approximation of pseudo-parabolic systems

E. Abreu,  E. Cuesta, A. Duran and W. Lambert.

Highlights:
The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are first studied. They include the weak formulation, well-posedness and existence of traveling wave solutions connecting two states, when the equations are considered as a variant of a conservation law. Then, the numerical approximation consists of a spectral approximation in space based on Legendre polynomials along with a temporal discretization with strong stability preserving (SSP) property. The convergence of the semidiscrete approximation is proved under suitable regularity conditions on the data. The choice of the temporal discretization is justified in order to guarantee the stability of the full discretization when dealing with nonsmooth initial conditions. A computational study explores the performance of the fully discrete scheme with regular and nonregular data.

Keywords: Pseudo-parabolic equations, spectral methods, error estimates, strong stability preserving methods, non-regular data.