Prerequisites: Linear Algebra and Applications, Analysis in Rn, Complex Analysis, Numerical Analysis, ODEs and PDEs.

Numerical analysis of elliptic partial differential equations. Numerical solution of the Laplace and Poisson equations via the Finite Difference Method (FDM), via the Finite Element Method (FEM), via Spectral Methods (e.g. Poisson’s Fast Solver) and via the Boundary Integrals Method (BIM). Numerical analysis of hyperbolic partial differential equations. Numerical solution of the convection equation (e.g. wave equation) using the MDF. Notions of consistency and stability. Stability analysis via the dispersion equation. Von Neumann stability analysis. Fourier analysis with grid functions, aliasing and the Poisson summation formula. Notions of numerical dissipation, numerical dispersion and modified differential equation. Lax equivalence theorem. Numerical solution of problems with discontinuity. Numerical solution of conservation laws. Numerical analysis of parabolic partial differential equations. Numerical solution of the diffusion equation (e.g. heat) by MDF and spectral methods.

References:
AMES, W. F. – Numerical Methods for Partial Differential Equations, 3rd. e ., Academic Press, 1992.
GOTTLIEB, D., ORSZAG, S. A. – Numerical Analysis of Spectral Methods, SIAM, 1977.
ISAACSON, E. and KELLER, H. – Analysis of Numerical Methods, Dover, 1966.
LE VEQUE, R. J. – Numerical Methods for Conservation Laws, Birkhäuser, 1992.
RICHTMEYER, R. D. and MORTON, K. W. – Difference Methods for Initial – Value Problems, Krieger Publ. Co., 2nd ed. 1967.
SMITH, G. D. – Numerical Solution of Partial Differential Equations, Finite Difference Methods, 3rd. ed., Oxford University Press, 1985.
STRIKWERDA, J. C. – Finite difference schemes and partial differential equations. 2nd ed Philadelphia: Society for Industrial and Applied Mathematics, 2004.
TREFETHEN, L. N. – Spectral Methods in MATLAB, SIAM, 2000.

* Basic syllabus. The teacher has the autonomy to make any changes.