GOALS

The Computational Fluid Dynamics, Mathematical Modeling & Numerical Methods research group has as its main goals:

• To integrate its participant members through joint research, departing from common interests in Computational Fluid Mechanics, Mathematical Modeling and Numerical Methods.
• To cooperate in the teaching load of graduate courses, supervision of dissertations and thesis, allowing for a solid background development for the undergraduate and graduate students.
• To estimulate the exchange of ideas with other research groups and institutions, from inside and outside of the University of Sao Paulo, and to collaborate for the diffusion of the scientific knowledge, to cooperate for the projects and other specialized services.

PARTICIPANT MEMBERS

Currently, this research group has six researchers, about one third of the Applied Mathematics Department.

Prof.Dr. Saulo Maciel Rabello de Barros (MS-5)
Dr. Rer. Nat.,  Universität Bonn

Prof.Dr. Alexandre Megiorin Roma (MS-3)
Ph.D., New York University - Courant Institute of Mathematical Sciences

Prof.Dr. Antonio Elias Fabris (MS-3)
Ph.D., University of East Anglia - School of Information Systems

Prof. Dra. Joyce da Silva Bevilacqua (MS-3)
Doctor, Universidade de Sao Paulo - Institute of Astronomy and Geophysics

Prof.Dr. Luis Carlos de Castro Santos (MS-3)
Ph.D., Georgia Institute of Technology - School of Aerospace Engineering

Prof.Dr.  Nelson Mugayar Kuhl (MS-3)
 Ph.D., New York University - Courant Institute of Mathematical Sciences

RESEARCH AREAS

The focus of this group is the numerical solution of differential equations coming from several areas and applications, as for example:

• Computational Fluid Mechanics
• To development numerical techniques to solve problems in aerodynamics, biofluid dynamics, physics of plasms, and weather numerical prediction. Regarding to these fields, the group researchs multigrid methods, self-adaptive refinement techniques, parallel algorithms, and inverse methods for compressible and incompressible flows.

• Mathematical Modeling and Numerical Analysis
• To study algorithms to solve numerically ordinary and partial differential equations. To investigate the modeling of systems in biology and industry. To study numerical techniques for solving systems of simultaneous equations, eigenvalue problems and for parallelizing numerical  algorithms.

ACADEMIC ACTIVITIES

Supervision of Students:  Currently, the participant members of this group supervise 10 graduate students enrolled in the scientific initiation program, 5 MSc and 6 Doctorate program students.

Mathematics Education: Investigation and developing new ways to teach mathematics, focusing its applications, employing modern technological resources such as audio and video, computer multimedia, and internet.

Graduate Courses: Among others, the most important graduate courses offered by this group are:

MAP5724 NUMERICAL RESOLUTION OF ELLIPTIC PARTIAL DIFFERENTIAL EQNS.

Contents: Second-order elliptic partial differential equations. Discretization methods: finite differences and finite elements. Classical relaxation methods: Gauss-Seidel and SOR. Gradient conjugate method; pre-conditioning. Direct methods. Fast Poisson solvers. Introduction to multigrid methods.

Bibliography: W. Hackbusch, Theorie und Numerik elliptischer Differentialgleichungen, Teubner, Stuttgart, 1986; W. Hackbusch, Multigrid methods and applications, Springer, 1985; K. Stuben, U. Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications. Springer, 1982. (Lecture Notes in Math. 960);  J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer, 1980.
 
 

MAP5725 NUMERICAL TREATMENT OF ORDINARY DIFFERENTIAL EQUATIONS

Contents: 1. A brief introduction to the theory of ordinary differential equations: existence, uniqueness, continuity, differentiability, and periodicity of solutions. First order systems. 2. Discretization methods: consistency, stability, and convergence. Forward step methods. Single and multiple step methods. Consistency. 3. Stability criteria. 4. Fixed point theorem. 5. Predictor-corrector methods. 6. Numerical stability of ODE wth initial conditions. Stiff ODEs. 7. Variable time stepping. Local truncation error. 8. Comparison between several methods.

Bibliography:  I.Q. Barros, Métodos numéricos em álgebra linear, vol. I., IMECC--UNICAMP, 1970;  L. Collatz, The numerical treatment of differential equations, Springer, 1960; P. Henrici, Discrete variational methods in ordinary differential equations, Wiley, 1962; J.D. Lambert, Computational methods in ordinary differential equations, Wiley, 1973; L.S. Pontriaguin,  Ordinary differential equations, Addison--Wesley, 1962;  H.J. Stetter, Analysis of discrete methods for ordinary differential equations, Springer, 1973.
 
 

MAP5726 INTRODUCTION TO THE COMPUTATIONAL FLUID DYNAMICS I:
                  INCOMPRESSIBLE FLUIDS

Contents: 1. A brief introduction to tensor algebra and analysis 2. Equations of motion 3. The solution in closed form for some particular examples 4. Some basic concepts of numerical analysis for partial differential equations 5. Non-dimensional form, similarity solutions, and Prandtl boundary layer equations 6. Generalized coordinates 7. Equivalent forms for the Navier-Stokes equations (conservative form, vorticity-stream,...) 8. Selection among several numerical methods: projection methods, vortex methods, spectral methods, finite element methods, etc.

 Bibliography:  Chorin, A.J.; Marsden, J.E., A mathematical introduction to fluid mechanics, Springer-Verlag; Gurtin, M.E.,An introduction to continuum mechanics, Academic Press; Serrin, J.,Mathmatical principles of classical fluid mechanics, Handbuch der Physik, VIII/1, Springer-Verlag; Strikwerda, J.C. Finite difference schemes and partial differential equations, Wadsworth and Brooks/Cole Mathematics Series; Peyret, R.; Taylor, T.D., Computational methods for fluid flow, Springer-Verlag.
 
 

 MAP5729 INTRODUCTION TO NUMERICAL ANALYSIS

Contents: 1. Resolution of linear systems: direct and iterative methods. 2. Resolution of non-linear equations: Richardson method, Newton. 3. Interpolation: polynomial (Lagrange and Hermite), splines. 4. Gaussian quadrature, Romberg method (based on extrapolation). 5.Numerical resolution of ordinary differential equations: initial value problems - single and multistep methods; contour problems; finite difference methods.

Bibliography:  J. Stoer, R. Bulirsch, Introduction to numerical analysis, Springer, Berlin, 1980; E. Isaacson, H.B. Keller, Analysis of numerical methods, Wiley, 1966.
 
 

MAP5745 FLUID MECHANICS

Contents: 1.Brief introduction to algebra and analysis. 2. Cinematics. 3. Momentum and mass. 4. Forces. 5. Constitutive hypothesis - inviscid fluids. 6. Change in the position of the observer - material invariance 7. Newtonian fluids - Navier--Stokes equations. 8. Finite elasticity. 9. Linear elasticity.

Bibliography:  M.E. Gurtin, An introduction to continuum mechanics, New York, Academic Press, 1981. 265p. (Mathematics in Science and Engineering Series)
 
 

MAP5822 MULTIGRID METHODS

Contents: 1. Introduction: basic concepts. 2. The model problem: Poisson equation on a rectangle.
Convergence analysis. Computational optimality.Variational formulation. 3. Elliptic systems. 4. Multigrid methods for finite elements. 5. Higher order discretizations, residual correction. 6.Multigrid e local refinement. FAC and AFAC. 7. Multigrid methods for hyperbolic and parabolic problems. 8. Parallelism. 9.Other topics as time permits.

Bibliography: A. Brandt, Multigrid techniques: 1984 guide with applications to fluid dynamics,GMD--Studie, 85 (St. Augustin), 1984; W. Hackbusch, Multigrid methods and applications, Berlin, Springer, 1985;  S. McCormick, Multilevel adaptive methods for partial differential equations, SIAM, Philadelphia, 1989; K. Stüber, U. Trottenberg, Multigrid methods: fundamental algorithms, model problem analysis and applications,  in: Hackbuich and Trottenberg (eds.), Berlin, Springer, 1982, (Lecture Notes in Mathematics, 960).