MATH 432: Numerical Differential Equations

MATH 432: Numerical Differential Equations - Spring 2006

Time: Tue & Thu 5:30pm-6:45pm, Yost 300
Instructor: Christophe Geuzaine (Office: Yost 213; Phone: 368 29 09; Email: cag32(at)case(dot)edu)
Office hours: Tue & Thu 12:00-1:00pm, Yost 213. If you want to see me, don't feel that you have to wait for an office hour. For shorter questions, just drop by at any time. For longer consultations, please make an appointment.
Textbook: A First Course in the Numerical Analysis of Differential Equations by Arieh Iserles (Cambridge Texts in Applied Mathematics, 1996). If you have an old copy of the book (printed before 2004), make sure to consult the erratum on Arieh Iserles' webpage.
Course description (from the catalog): Numerical solution of differential equations for scientists and engineers. Solution of ordinary differential equations by multistep and single step methods. Stability, consistency, and convergence. Stiff equations. Finite difference schemes. Introduction to the finite element method. Introduction to multigrid techniques. The diffusion equation: numerical schemes and stability analysis. Introduction to hyperbolic equations. Matlab will be used in this course.
Grading: There will be an in-class midterm exam and a final project. The midterm will determine 30% of your grade, and the final project will determine 40%. The remaining 30% will be based on homework assignments.
Homework: Homework will be assigned and collected every week. It is important that you keep up with the work: do the homework as it is assigned, and ask questions right away if you find there is something you don't understand. The easiest way to do poorly in this class is to get behind on the homework assignments.
Midterm: Thu 03/09

Schedule and Assignments

	Chapter	Homework
Wed 01/18	administrative	.
Tue 01/24	1.1, 1.2: Euler's method, convergence	.
Thu 01/26	1.2-1.4: local/global truncation errors, trapezoidal rule, theta method	Assignment 1
Tue 01/31	2.1, A.2.2: multistep intro, polynomial interpolation	.
Thu 02/02	2.1, 2.2, 5.1: Adams-Bashforth, Adams-Moulton, predictor-corrector, order of multistep methods	Assignment 2
Tue 02/07	2.2, 2.3: convergence of multistep methods, Dahlquist equivalence theorem	.
Thu 02/09	2.3, 3.1: BDF schemes, Gaussian quadrature	Assignment 3
Tue 02/14	3.1, 3.2: explicit Runge-Kutta methods	.
Thu 02/16	3.3, 4.2: implicit Runge-Kutta methods, zero stability, absolute stability	Assignment 4
Tue 02/21	4.2: linear stability domain	.
Thu 02/23	4.1, 4.3, 4.4: stiff ODEs, A-stability of multistep and Runge-Kutta methods (Matlab demos: Stiff1.m, Stiff1b.m, Stiff2.m, EulerSystem.m)	Assignment 5
Tue 02/28	5.1, 5.2, 5.3: error control, Milne device, Richardson extrapolation, embedded Runge-Kutta methods	.
Thu 03/02	6.1, 6.2: functional iteration, Newton-Raphson, Modified Newton-Raphson and Quasi-Newton iteration	.
Tue 03/07	7.1: ODE review, PDE intro	.
Thu 03/09	midterm exam	.
Tue 03/14	no class (spring break)	.
Thu 03/16	no class (spring break)	.
Tue 03/21	x.x: midterm solutions, shooting methods for two-point boundary value problems	.
Thu 03/23	H6.1, H6.2: finite differences for two-point boundary value problems	.
Tue 03/28	7.2, 7.3, H6.6, 9.x, 10.x: finite differences for elliptic PDEs, direct and iterative sparse solvers	.
Thu 03/30	x.x, 8.1: collocation methods, Ritz-Galerkin methods	.
Tue 04/04	8.1, 8.2, 8.3: finite elements for elliptic PDEs	.
Thu 04/06	8.2, 8.3: optimality of Ritz-Galerkin, variational setting, examples (GetDP demo: elliptic.geo, elliptic.pro)	.
Tue 04/11	13.1, 13.2, H6.3, H6.4: numerical methods for parabolic PDEs, convergence	.
Thu 04/13	13.3, 13.4, H6.3: numerical methods for parabolic PDEs, examples, stability (GetDP demo: parabolic.geo, parabolic.pro)	.
Tue 04/18	13.5, 14.1, H6.4: stability analysis by Fourier techniques, intro to hyperbolic PDEs (GetDP demo: hyperbolic.geo, hyperbolic.pro)	.
Thu 04/20	14.1, 14.2, 14.3, 14.5: numerical methods for hyperbolic PDEs	Draft presentations due
Tue 04/25	final project presentations	.
Thu 04/27	last class, final project presentations	Final projects due

Useful links

Matlab is available via the software center. (Here are some Matlab tutorials.)
In addition to the texbook, we may use some material from Nick Trefethen's online text: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations. (And even if we don't, it's still a very good read...)