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Study of partial differential algebraic equations of hyperbolic-parabolic dominance with relaxation: theory, numerics and applications

Abstract

Partial differential equations (PDEs) and algebraic equations (DAEs) arise in numerous mathematical models in science and engineering. In recent years, there has been a growing interest of the study of coupled systems of PDEs and DAEs in a wide range of applications, for instance, in sensitivity analysis, chemical process control, parameter estimation, data assimilation, control of PDEs, uncertainty analysis, porous media, experimental design, Materials and Design, Electrical Network Modeling, Biology, just to name a few topics of current events. In this project, we interested in studying highly nonlinear systems of coupled problems related to PDEs and DAEs. We are particulary interest in the study of PDEs and DAEs with relaxation terms of hyperbolic-parabolic dominance. This set of equations are used to describe accurately models that exhibit very complex physics, for instance, in problems with thermal or (geochemical) chemical flows. These flows are modeled using rigorous and fundamental thermodynamic laws and, since there are chemical reactions, the chemical laws are also important. Physical and chemical laws give origin to algebraic equations. States satisfying algebraic equations generate a surface called equilibrium surface. For many non-linear models it is impossible to obtain a set of variables to parameterize or simplify this equilibrium surface and we need to solve the full coupled system of equations formed, namely, we need to solve the complete system of PDEs and DAEs subject to pertinent boundary and initial conditions, or simply PDAEs. Moreover, we are interested to connect the theory of PDAEs with the theory of relaxation equations by adding source terms in PDAEs supported on solid grounds of fundamental thermodynamic laws and mathematics. Relaxation systems of equations are very important because they give a natural way to obtain physical solutions from weak solutions without the necessity of defining additional entropy criteria. We intend to obtain analytical results and novel numerical balancing discretization methods for this very large class of PDAEs with hyperbolic-parabolic dominance and to apply both theory and numerics to solve complex thermal and chemical problems for multiphase transport systems as well as modeling phase change in problems with phase transition with relaxation. (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; FERRAZ, PAOLA; VIEIRA, JARDEL. Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media. Journal of Computational Physics, v. 411, . (19/20991-8)
ABREU, EDUARDO; FRANCOIS, JEAN; LAMBERT, WANDERSON; PEREZ, JOHN. A Class of Positive Semi-discrete Lagrangian-Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws. JOURNAL OF SCIENTIFIC COMPUTING, v. 90, n. 1, . (19/20991-8)
DE LA CRUZ, RICHARD; SANTOS, MARCELO; ABREU, EDUARDO. Interaction of delta shock waves for a nonsymmetric Keyfitz-Kranzer system of conservation laws. MONATSHEFTE FUR MATHEMATIK, v. 194, n. 4, . (16/19502-4, 19/20991-8)
ABREU, EDUARDO; DE LA CRUZ, RICHARD; LAMBERT, WANDERSON. iemann problems and delta-shock solutions for a Keyfitz-Kranzer system with a forcing ter. Journal of Mathematical Analysis and Applications, v. 502, n. 2, . (19/20991-8)
LAMBERT, WANDERSON; ALVAREZ, AMAURY; LEDOINO, ISMAEL; TADEU, DUILIO; MARCHESIN, DAN; BRUINING, JOHANNES. Mathematics and Numerics for Balance Partial Differential-Algebraic Equations (PDAEs). JOURNAL OF SCIENTIFIC COMPUTING, v. 84, n. 2, . (19/20991-8)
ABREU, EDUARDO; DIAZ, CIRO; GALVIS, JUAN; PEREZ, JOHN. ON THE CONSERVATION PROPERTIES IN MULTIPLE SCALE COUPLING AND SIMULATION FOR DARCY FLOW WITH HYPERBOLIC-TRANSPORT IN COMPLEX FLOWS. MULTISCALE MODELING & SIMULATION, v. 18, n. 4, p. 1375-1408, . (19/20991-8)
ABREU, E.; MATOS, V.; PEREZ, J.; RODRIGUEZ-BERMUDEZ, P.. A Class of Lagrangian-Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms. JOURNAL OF SCIENTIFIC COMPUTING, v. 86, n. 1, . (19/20991-8)

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