Numerical Solution of Differential Equations

Math 686-001 (Spring 2024)

Detailed Syllabus

The following table contains the preliminary schedule for the course. Changes to the topics are possible, based on student input. This page will be updated regularly throughout the semester.

Week	Dates	Pages		Book Sections
1	01/16 - 01/19	1-14	Introduction & Overview
			I. Basic Methods for Ordinary Differential Equations
			1. Initial Value Problems for ODEs	[I] 1.1, [SW] 1.1.1, 1.1.2
			2. Time-Stepping for Initial Value Problems	[I] 1.2, [SW] 3.3.1
2	01/22 - 01/26	15-25	3. Euler’s Method and Beyond	[I] 1.2, 1.3, 1.4, [SW] 3.3.1
			4. Multistep Methods	[I] 2.1, 2.2, 2.3
3	01/29 - 02/02	26-31	5. Gaussian Quadrature	[I] 3.1
	02/01		No class!
4	02/05 - 02/09	32-41	6. Runge-Kutta Methods	[I] 3.2, 3.3
			7. Collocation Methods	[I] 3.4
			II. Qualitative Study of ODE Methods
5	02/12 - 02/16	42-53	1. Stiff Differential Equations	[I] 4.1
			2. Stability of Numerical Methods	[I] 4.2
			3. Stability of Runge-Kutta Methods	[I] 4.3
6	02/19 - 02/23	54-64	4. A-Stability of Multistep Methods	[I] 4.4
			5. Error and Stepsize Control	[I] 6.1, 6.2
7	02/26 - 03/01	65-75	6. Embedded Runge-Kutta Methods	[I] 6.3
			7. Geometric Numerical Integration	[I] 5.1, 5.2, 5.4
8	03/04 - 03/08		No class! (Spring Break)
9	03/11 - 03/15	76-85	III. Approximation of Spatial Derivatives
			1. Derivatives via Finite Differences	[SW] 3.1.1, 3.1.3
			2. Spectral Interpolation and Differentiation	[SW] 3.4.1
10	03/18 - 03/22	86-90	3. Cosine Sums and the Discrete Cosine Transform	[SW] 3.4.1, 3.5.1
	03/19		No class!
11	03/25 - 03/29	91-104	4. Chebyshev Differentiation	[SW] 3.4.2, 3.5.3
			5. Two-Dimensional Spectral Methods	[SW] 3.4.3
			IV. Stationary Partial Differential Equations
12	04/01 - 04/05	105-116	1. Review of Linear Elliptic Equations	[SW] 4.1.1
			2. Finite Differences in One Dimension	[SW] 4.2.1
13	04/08 - 04/12	117-132	3. Theoretical Aspects of Finite Differences	[SW] 4.2.3
			4. Rectangular Domains and Sparse Matrices	[SW] 4.2.2
14	04/15 - 04/19	133-144	5. Chebyshev Methods for Elliptic Equations	[SW] 4.3.2
			6. Nonlinear Elliptic Problems	[SW] 5.1.1, 5.1.2
			7. Spurious Solutions in Nonlinear Problems	[SW] 5.1.1, 5.3.1
			V. Partial Differential Equations of Evolution
15	04/22 - 04/26	145-170	1. The Diffusion Equation	[I] 16.1, 16.2, 16.3, [SW] 6.1.1, 6.1.2
			2. The Wave Equation	[I] 17.1, 17.2, 17.4
			3. Nonlinear Parabolic Equations	[SW] 6.1.3
16	04/30		Final Presentations! (6:00-8:00pm)	Ratcliffe, Blauvelt, Pryor, Ro
	05/02		Final Presentations! (6:00-7:30pm)	Kinseth, Chisholm, Althobaiti

For the course, I will draw material from the following books:

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009 (2nd edition).
E. Sander, T. Wanner, Theory and Numerics of Partial Differential Equations, Book Manuscript, in preparation, 2024.
E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, Springer, 2000 (2nd edition).
E. Hairer, G. Wanner, Solving Ordinary Differential Equations II, Springer, 2002 (2nd edition).
E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration, Springer, 2006 (2nd edition).
J.C. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley, 2003.
L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, 1994.