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Conservation laws, balance laws and related PDEs with discontinuous and nonlocal fluxes in applied sciences: numerical analysis, theory and applications


Compositional and reactive flows, relaxation and bistability interactions, and nonlocal effects along with continuous/discrete random modeling in applied sciences are of paramount importance in mathematical models involving conservation laws, balance laws, related non conventional partial differential equations (PDEs) and vice-versa. The behaviour of every physical phenomena, with dynamics for a wide range of space-time scales -- transport processes that span time scales of picoseconds to millennia and spatial dimensions of angstroms to kilometers or astronomical scale -- can be modelled by such differential mathematical models. Such objects are an essential tool in modeling, analysis, and prediction of numerous physical, biological, chemical and economic systems in which the applied mathematics plays a large role. Their unifying theme is the theory of solutions for nonlinear differential models linked to approximation theory for numerical analysis, supplemented by representative, accurate, fast and efficient numerical simulations. In each topic of this research project, nonlinear problems involving diverse differential equations of mixed type will be posed and corresponding mathematical and numerical techniques for their solution will be devised. As a result, new mathematical and physical insight will be gained by investigating important nonlinear dynamics of differential models, effective nonlinear techniques will be developed, and the correct mathematical frameworks in which to pose and discuss these problems will be pursued. For instance, nonlinear PDEs possess solutions exhibiting singularities, oscillations, and/or concentration effects, which in the real world are reflected in the appearance of shock waves, turbulence, material defects, etc. This immediately poses very fundamental questions such as what is the nature and effect of noise on singularities and can the solution be continued past singularity formation and also in the late time (relaxation) behavior ? Questions like these are intimately tied to several core issues, such as understanding what we actually mean by a (notion of) solution. Thus, the development of theories about existence, uniqueness and stability of solutions is linked to the construction of numerical algorithms and thus, in core of applied mathematics. Over the last few decades these questions have been given satisfactory answers for several classes of deterministic nonlinear partial differential equations. However, the situation is dramatically different for conservation laws, balance laws and related PDEs with discontinuous and nonlocal fluxes and subject to relaxation linked to the late time behavior. An overall goal of the project is to develop concepts and techniques for the analysis of phenomena modeled by such nonlinear models with solutions possessing low regularity (e.g., shock waves). It will involve several branches of mathematics, including partial differential equations, functional and numerical analysis and simulations and using a range of techniques such as entropy and viscosity analysis, a priori estimates and a novel concept of weak asymptotic approximation solutions. Furthermore, since the design of efficient numerical schemes hinges on the understanding of the underlying mathematical structure and pattern we have a unified approach involving numerical analysis, theory and applications linked to applied mathematics issues. (AU)

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Scientific publications (7)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ABREU, EDUARDO; DIAZ, CIRO; GALVIS, JUAN; SARKIS, MARCUS. On high-order conservative finite element methods. COMPUTERS & MATHEMATICS WITH APPLICATIONS, v. 75, n. 6, SI, p. 1852-1867, . (16/23374-1)
ABREU, EDUARDO; BUSTOS, ABEL; FERRAZ, PAOLA; LAMBERT, WANDERSON. A Relaxation Projection Analytical-Numerical Approach in Hysteretic Two-Phase Flows in Porous Media. JOURNAL OF SCIENTIFIC COMPUTING, v. 79, n. 3, p. 1936-1980, . (16/23374-1)
ABREU, EDUARDO; PEREZ, JOHN; SANTO, ARTHUR. Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws. UIS INGENIERIAS, v. 17, n. 1, p. 191-200, . (16/23374-1)
ABREU, EDUARDO; PEREZ, JOHN. A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications. COMPUTERS & MATHEMATICS WITH APPLICATIONS, v. 77, n. 9, p. 2310-2336, . (16/23374-1)
ABREU, EDUARDO; DIAZ, CIRO; GALVIS, JUAN; SARKIS, MARCUS. On high-order conservative finite element methods. COMPUTERS & MATHEMATICS WITH APPLICATIONS, v. 75, n. 6, p. 16-pg., . (16/23374-1)
SANTOS, L. G. C.; MANZANARES-FILHO, N.; MENON, G. J.; ABREU, E.. Comparing RBF-FD approximations based on stabilized Gaussians and on polyharmonic splines with polynomials. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, v. 115, n. 4, p. 462-500, . (16/23374-1)
ABREU, EDUARDO; DIAZ, CIRO; GALVIS, JUAN. A convergence analysis of Generalized Multiscale Finite Element Methods. Journal of Computational Physics, v. 396, p. 303-324, . (16/23374-1)

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