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Prof. L. Zovatto
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1.
Boundary conditions:
Essential or Dirichlet boundary conditions
Natural or Neumann boundary conditions
Mixed boundary conditions and initial conditions
The Cauchy problem
Solution of the D’Alembert equation
2.
Finite Elements Method (FEM)
The finite elements
Form function
Areolar coordinates
Nodal form functions
Galerkin method
Application of the Galerkin method
Assembly of the linear system
Gradients of the form functions
Finite Differences (FD)
The differential operator in discrete form
FD schemes
Boundary conditions
Stretched grids
Finite Volumes Method (FVM)
Evaluation of terms appearing in the integral equation
Assembly of the linear system
Cell Elements Method (CEM)
Poisson equation: circumcentric polygons
Poisson equation: barycentric polygons
Introduction of essential boundary conditions
Type A direct method
Type B direct method
Other methods
3.
Explicit methods
Euler method
Heun method

Runge-Kutta methods
Implicit methods
Predictor-corrector methods
Runge-Kutta schemes for systems of differential equations
Pseudo-exact solutions
Semi-implicit methods
Convergence
Stability analysis
Convergence order
Schemes for the Navier-Stokes equations
Consistency analysis
Fractional step in primitive variables
Advancement in vorticity-streamfunction formulation
1.
M. Abramowiz, I. Stegun HandBook of mathematical function. Dover Publication, 1970
2.
C. A. Brebbia. The boundary element method for engineers. Pentech Press,1980.
3.
C.Canuto, M.Hussaini, A.Quarteroni and T.Zang, Spectral Methods in Fluid Dynamics Springer
Verlag 1988
4.
Flecther C.A.J. Computational Techniques for Fluid Dynamics. Vol.1,2 Springer-Verlag, 1988.
5.
Hirsch C. Numerical Compuration of Internal And External Flows. Vol.1,2 John Wiley & Sons,
1996.
6.
D. Greenspan, V. Casulli Numerical Analysis for Applied Mathematics, Science, and Engineering,
Addison Wesley Publishing Company, 1993.
7.
W.G. Gray, D.R. Time-stepping schemes for finite element tidal model computations.. Advances in
Water resources, Vol.1 , No. 2,1977.
8.
A.R. Mitchell, R.Wait . The finite element method in partial differenzial equation. Wiley 1978.
9.
A.R. Mitchell, D.F. Griffiths The finite difference method in partila differenzial equation. Wiley
1980.
10.
Patrick J. Roache Fundamentals of Computational Fluid Dynamics. Hermosa publisher 1998.
11.
Rao. The Finite Element method In Enginnering A. Wheaton & Co. Ltd, 1982.
12.
H.K Versteeg, W. Malalasekera. An introduction to Computational Fluid Dynamics. The finite volume method. Longman Scientific Tecnichal, 1995.
13.
E. Tonti. A Direct Discrete Formulation of Field Laws: The Cell Method. Computer Modelling in
Engineering and Science, CMES, Vol. 2, No. 2, pp. 237-258, 2001.
14.
E. F. Toro. Shock-capturing methods for free-surface shallow flows. Wiley, 2001.
15.
J. H. Ferziger, M. Peri´c. Computational Methods for Fluid Dynamics. Springer , 1999.
16.
Zinkiwiecz. The Finite Element method. Mc Graw Hill, 1990.