# Numerical Methods for Partial Differential Equations

**Prerequisites: **Linear Algebra and Applications, Analysis on Rn, Complex Analysis, Numerical Analysis, EDO’s and EDP’s.

Numerical analysis of hyperbolic partial differential equations. Numerical solution of convection equations (eg the wave equation) through the Finite Difference Method (FDM). Notions of consistency and stability. Stability analysis via the dispersion equation. The von Neumann stability analysis. Fourier analysis with grid functions, “aliasing” and the Poisson summation formula. Notions of numerical dissipation, numerical dispersion and the modified differential equation. Lax equivalence theorem. Numerical solution of problems with discontinuities. Numerical solution of Conservation Laws.

Numerical Analysis of parabolic differential equations. Numerical solution of the diffusion equation (eg heat) by the FDM and spectral methods. Numerical analysis of elliptic partial differential equations. Numerical solution of Laplace and Poisson equation via the finite difference method (FDM). If time permits notions on the numerical solution via a Spectral Methods (Fast Poisson solver) and the Boundary Integral Method (BEM).

**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. e KELLER, H. – Analysis of Numerical Methods, Dover, 1966.

LE VEQUE, R. J. – Numerical Methods for Conservation Laws, Birkhäuser, 1992.

RICHTMEYER, R. D. e 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.

* Standard program. The teacher has the autonomy to make any changes