Instructor: Dr. I. Oprea http://www.math.colostate.edu/~juliana

MWF 11:00 - 11:50, Room: Engineering E203

Analytic solutions exist only for the most elementary partial differential equations (PDEs); the rest must be tackled with numerical methods. This course will cover numerical solution of PDEs: the method of lines, finite differences, finite element and spectral methods, to an extent necessary for successful numerical modeling of physical phenomena. We shall construct and characterize the behavior of computational methods for PDEs based on the physical meaning and the mathematical analysis of the underlying continuous equations

This course is an entry-level graduate course. The intended audience is graduate students and advanced undergraduates. The design philosophy of the course is to cover the mathematical theory applied to the simplest examples so as to minimize technical details, yet provide a strong foundation that students can carry back to their particular fields

__Coursework__

The course work consists
of a mixture of mathematical problem sets and computational projects.
The computational projects emphasize experimentation with mathematical
issues such as convergence, stability and accuracy.

__Prerequisites__

The course assumes
familiarity with basic linear algebra and ordinary differential
equations. Exposure to solutions of the classic models in partial
differential equations is a plus. The use of MATLAB is strongly
encouraged.

__Textbooks__

1. Computational Differential
Equations, K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Cambridge
University Press, 1996

2. Numerical Solution of Partial Differential Equations: An
Introduction, K. Morton and D. Mayers, Cambridge University Press, 2-nd
edition

Topics to be Covered

- Interpolation theory, Numerical quadrature, The need for numerical solutions of differential equations

2. Spectral Methods

- an overview

- Models involving conservation of heat, behavior of solutions

- Two-point boundary value problems and the Laplace and Poisson equations

- The variational formulation and the Galerkin finite element method

- Convergence in the energy norm, a priori convergence, order of convergence

- Quadrature in the finite element method and the finite difference method

- Brief overview of complications that can occur in realistic models

- Explicit and implicit method of lines using finite elements in space and finite differences in time

- Numerical stability, stiffness and dissipativity, convergence

- The transport equation and wave equations, characteristics and the transport of information, behavior of solutions

- Finite difference schemes, consistency

- Stability, dissipativity, dispersion, the CFL condition, convergence