The Computational Fluid Dynamics, Mathematical Modeling & Numerical Methods research group has as its main goals:
Currently, this research group has six researchers, about one third of the Applied Mathematics Department.
The focus of this group is the numerical solution of differential equations coming from several areas and applications, as for example:
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,
numerical treatment of differential equations, Springer, 1960; P. Henrici,
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:
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,
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
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,
methods and applications, Berlin, Springer, 1985; S. McCormick,
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).