Syllabus academic year 2010/2011
(Created 2010-07-25.)
NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONSFMNN10
Credits: 8. Grading scale: TH. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course will be given in English on demand. FMNN10 overlaps following cours/es: FMN011, FMN041, FMN050, FMN081, FMN130 and FMNF01. Compulsory for: F3, Pi3. Optional for: I4. Course coordinator: Gustaf Söderlind, Gustaf.Soderlind@na.lu.se and Anders Holst, Anders.Holst@math.lth.se, Numerical Analysis. Recommended prerequisits: FMA420 Linear Algebra, FMA430 Calculus in Severable Variables, FMA021/FMAF15 Applied Mathematics. Assessment: The grade is based on homework assignments and a written exam. Home page: http://www.maths.lth.se/na/courses/FMNN10.

Aim
The aim of the course is to teach computational methods for solving ordinary and partial differential equations. This includes the construction, application and analysis of basic computational algorithms. Problem solving by computers is a central part of the course.

Knowledge and understanding
For a passing grade the student must

- be able to discretize ordinary and partial differential equations. Moreover, students have to independently implement and to apply such algorithms.

Skills and abilities
For a passing grade the student must

- be able to independently select and apply computational algorithms.

- be able to evaluate both accuracy and relevance of numerical results.

- report solutions to problems and numerical results in written form.

Judgement and approach
For a passing grade the student must

- write a logically well structured report in suitable terminology on the construction of basic mathematical models and algorithms.

- write an algorithmically well structured report in suitable terminology on the numerical solution of a mathematical problem.

Contents
Methods for time integration: Euler’s method, the trapezoidal rule. Multistep methods: Adams methods, backward differentiation formulae. Explicit and implicit Runge-Kutta methods. Error analysis, stability and convergence. Stiff problems and A-stability. Error control and adaptivity. The Poisson equation: Finite differences and the finite element method. Elliptic, parabolic and hyperbolic problems. Time dependent PDEs: Numerical schemes for the diffusion equation. Introduction to difference methods for conservation laws.

Literature
Iserles, A: Numerical analysis of differential equations. Cambridge University Press, 1996, ISBN 0-521-55655-4.
Edsberg, L: An Introduction to Modeling and Computing for Differential Equations. Wiley 2008. ISBN 0470270853, 9780470270851.