NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS FMN130

Higher education credits: 7,5. Grading scale: TH. Level: A (Second level). Language of instruction: The course will be given in English on demand. FMN130 overlap following cours/es: FMN011, FMN041, FMN050, FMN081, FMN011, FMN041, FMN050 och FMN081. Compulsory for: Pi3. Alternative for: B3, K3. Optional for: F4, I4, K4p. Course coordinator: Gustaf Söderlind, Gustaf.Soderlind@na.lu.se, Numerisk analys. Recommended prerequisits: Linear algebra, calculus, applied mathematics. The course might be cancelled if the numer of applicants is less than 10. Assessment: The grade is based on homework assignments and a written exam. Home page: http://www.maths.lth.se/na/courses/FMN130/.

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. Problemsolving 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, that is to construct computable approximations. Moreover, students have to implement and to apply such algorithms independently

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

Judgement and approach
For a passing grade the student must

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

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

- write a 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. Differential algebraic systems. The Poisson equation: Finite differences and the finite element method, multigrid. Time dependent PDEs: Numerical schemes for the diffusion equation. An introduction to finite volume schemes for conservation laws.

Literature
Iserles, A: Numerical analysis of differential equations. Cambridge University Press, 1996, ISBN 0-521-55655-4.
Tveito, A. and Winther, R.: Introduction to partial differential equations. A computational approach. Springer 1998, ISBN 0-387-98327-9