@article {ABREU2024107967, title = {A relaxation approach to modeling properties of hyperbolic{\textendash}parabolic type models}, journal = {Communications in Nonlinear Science and Numerical Simulation}, volume = {133}, year = {2024}, pages = {107967}, abstract = {

In this work, we propose a novel relaxation modeling approach for partial differential equations (PDEs) involving convective and diffusive terms. We reformulate the original convection{\textendash}diffusion problem as a system of hyperbolic equations coupled with relaxation terms. In contrast to existing literature on relaxation modeling, where the solution of the reformulated problem converges to certain types of equations in the diffusive limit, our formalism treats the augmented problem as a system of coupled hyperbolic equations with relaxation acting on both the convective flux and the source term. Furthermore, we demonstrate that the new system of equations satisfies Liu{\textquoteright}s sub-characteristic condition. To verify the robustness of our proposed approach, we perform numerical experiments on various important models, including nonlinear convection{\textendash}diffusion problems with discontinuous coefficients. The results show the promising potential of our relaxation modeling approach for both pure and applied mathematical sciences, with applications in different models and areas.

}, keywords = {Convection{\textendash}diffusion problems, Discontinuous coefficient in space, Discontinuous flux function, Modeling using PDEs, Numerical validation of models, Relaxation, Sub-characteristic condition}, issn = {1007-5704}, doi = {https://doi.org/10.1016/j.cnsns.2024.107967}, url = {https://www.sciencedirect.com/science/article/pii/S1007570424001539}, author = {Abreu, Eduardo and Arthur Esp{\'\i}rito Santo and Lambert, Wanderson and P{\'e}rez, John} } @article {vasques2024acesso, title = {ACESSO DE ESTUDANTES AUTISTAS {\`A} EDUCA{\c{C}}{\~A}O B{\'A}SICA NO BRASIL: uma an{\'a}lise a partir do censo escolar}, journal = {Curr{\'{\i}culo sem Fronteiras}, volume = {24}, year = {2024}, note = {n/a}, pages = {e1134}, abstract = {n/a}, author = {Vasques, Carla K and do Esp{\'{\i}rito Santo, Arthur Miranda and Meletti, Silvia M{\'a}rcia Ferreira} } @article {https://doi.org/10.1002/num.22972, title = {Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian{\textendash}Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems}, journal = {Numerical Methods for Partial Differential Equations}, volume = {39}, number = {3}, year = {2023}, pages = {2400-2443}, abstract = {

Abstract We design and implement an effective fully discrete Lagrangian{\textendash}Eulerian scheme for a class of scalar, local and nonlocal models, and systems of hyperbolic problems in 1D. We propose statements, via a weak asymptotic analysis, which include existence, uniqueness, regularity, and numerical approximations of entropy-weak solutions computed with the scheme for the corresponding nonlinear initial value problem for the local scalar case. We study both convergence and weak bounded variation (BV) properties of the scheme to the entropy solution (for the local and scalar case) in the sense of Kruzhkov. The approach is based on the improved concept of no-flow curves, as introduced by the authors, and we highlight the strengths of the method: (i) the scheme for systems of hyperbolic problems does not require computation of the eigenvalues (exact or approximate) either to the numerical flux function or the weak CFL stability condition (wCFL) and (ii) we prove the properties: positivity-preserving, total variation nonincreasing, and maximum principle subject to the wCFL. We present numerical experiments to evaluate the shock capturing capabilities of the scheme in resolving the main features for hyperbolic problems: shock waves, rarefaction waves, contact discontinuities, positivity-preserving properties, and nonlinear wave formations and interactions.

}, keywords = {fully discrete Lagrangian{\textendash}Eulerian scheme, Kruzhkov entropy condition, maximum principle, no-flow curves, positivity-preserving, total variation nonincreasing, weak asymptotic analysis, weak bounded variation}, doi = {https://doi.org/10.1002/num.22972}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/num.22972}, author = {Abreu, Eduardo and Arthur Esp{\'\i}rito Santo and Lambert, Wanderson and P{\'e}rez, John} } @article {ABREU2023111681, title = {Recursive formulation and parallel implementation of multiscale mixed methods}, journal = {Journal of Computational Physics}, volume = {473}, year = {2023}, pages = {111681}, abstract = {

Multiscale methods for second order elliptic equations based on non-overlapping domain decomposition schemes have great potential to take advantage of multi-core, state-of-the-art parallel computers. These methods typically involve solving local boundary value problems followed by the solution of a global interface problem. Known iterative procedures for the solution of the interface problem have typically slow convergence, increasing the overall cost of the multiscale solver. To overcome this problem we develop a scalable recursive solution method for such interface problem that replaces the global problem by a family of small interface systems associated with adjacent subdomains, in a hierarchy of nested subdomains. Then, we propose a novel parallel algorithm to implement our recursive formulation in multi-core devices using the Multiscale Robin Coupled Method by Guiraldello et al. (2018) [26], that can be seen as a generalization of several multiscale mixed methods. Through several numerical studies we show that the new algorithm is very fast and exhibits excellent strong and weak scalability. We consider very large problems, that can have billions of discretization cells, motivated by the numerical simulation of subsurface flows.

}, keywords = {Darcy{\textquoteright}s law, Domain decomposition, Fluid dynamics in porous media, Mixed finite elements, Parallelization, Recursive Multiscale Robin Coupled Method}, issn = {0021-9991}, doi = {https://doi.org/10.1016/j.jcp.2022.111681}, url = {https://www.sciencedirect.com/science/article/pii/S0021999122007446}, author = {E. Abreu and P. Ferraz and A.M. {Esp{\'\i}rito Santo} and F. Pereira and L.G.C. Santos and F.S. Sousa} } @conference {abreu2015lagrangian, title = {A weak asymptotic solution analysis for a Lagrangian-Eulerian scheme for scalar hyperbolic conservation laws}, booktitle = {Proceedings of the Seventeenth International Conference on Hyperbolic Problems}, year = {2020}, pages = {223{\textendash}230}, address = {University Park, Pennsylvannia, United States of America}, author = {Abreu, Eduardo and Lambert, Wanderson and Perez, John and Santo, Arthur} } @conference {abreu2018conservative, title = {A conservative Lagrangian-Eulerian finite volume approximation method for balance law problems}, booktitle = {Proceeding Series of the Brazilian Society of Computational and Applied Mathematics}, volume = {6}, number = {1}, year = {2018}, address = {S{\~a}o Jos{\'e} dos Campos, S{\~a}o Paulo - Brazil}, author = {Abreu, Eduardo and P{\'e}rez, John and Santo, Arthur} } @article {abreu2018lagrangian, title = {Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws}, journal = {Revista UIS Ingenier{\'\i}as}, volume = {17}, number = {1}, year = {2018}, pages = {191{\textendash}200}, publisher = {Universidad Industrial de Santander}, author = {Abreu, Eduardo and Perez, Jhon and Santo, Arthur} } @article {abreu2017new, title = {A new finite volume approach for transport models and related applications with balancing source terms}, journal = {Mathematics and Computers in Simulation}, volume = {137}, year = {2017}, pages = {2{\textendash}28}, publisher = {Elsevier}, author = {Abreu, Eduardo and Lambert, Wanderson and Perez, J and Santo, A} } @conference {abreu2017solving, title = {Solving hyperbolic conservation laws by using Lagrangian-Eulerian approach}, booktitle = {Proceeding Series of the Brazilian Society of Computational and Applied Mathematics}, volume = {5}, number = {1}, year = {2017}, address = {Gramado, Rio grande do Sul - Brazil}, author = {Abreu, Eduardo and Perez, John and Santo, Arthur} } @conference {abreu2015lagrangian, title = {A Lagrangian-Eulerian algorithm for solving hyperbolic conservation laws with applications}, booktitle = {Proceedings of the 6th International Conference on Approximation Methods and Numerical Modelling in Environment and Natural Resources (MAMERN VI)}, year = {2015}, pages = {599{\textendash}617}, address = {Pau - France}, author = {Abreu, Eduardo and Lambert, Wanderson and Perez, John and Santo, Arthur} }