Overview

Description

CALENDAR DESCRIPTION:

Analysis and application of numerical methods for solving partial differential equations. Potential topics include finite difference methods, spectral methods, finite element methods, and multi-level/multi-grid methods.

COURSE DETAILS:

This course provides an introductory overview of the numerical analysis of differential equations, covering the derivation and theoretical background for the methods as well as their computational implementation.  A tentative course outline is as follows:  

Finite differencing (2.5 weeks): explicit and implicit discretizations; local and global errors, consistency, stability and convergence; treatment of boundary conditions; higher-dimensional elliptic problems, Poisson’s equation  

Spectral methods (2.5 weeks): discrete Fourier transform and FFT, spectral accuracy; Fourier and Chebyshev spectral methods  

Finite elements (2.5 weeks): two-point boundary value problems, introduction to theory, Poisson’s equation  

Initial value problems (2 weeks): Runge-Kutta and multistep methods, zero-stability and absolute stability, stiffness  

Parabolic equations (1 week): heat equation, discretizations; accuracy and stability, von Neumann error analysis, Lax equivalence theorem  

Hyperbolic equations (1.5 weeks): advection equation; leapfrog, Lax-Friedrichs, Lax-Wendroff and upwind schemes; CFL condition; modified equations  

Mixed and nonlinear PDEs (1 week): fractional step methods, IMEX methods, pseudo-spectral methods, exponential time differencing

Grading

Materials

REQUIRED READING:

RECOMMENDED READING: