Numeriska metoder för differentialekvationer

**Valid from:** Autumn 2019**Decided by:** Professor Thomas Johansson**Date of establishment:** 2019-10-08

**Division:** Numerical Analysis**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course codes:** FMNN10, NUMN12**Teaching language:** English

The aim of the course is to teach computational methods for solving both ordinary and partial differential equations. This includes the construction, application and analysis of basic computational algorithms for approximate solution on a computer of initial value, boundary value and eigenvalue problems for ordinary differential equations, and for partial differential equations in one space and one time dimension. Independent problem solving using computers is a central part of the course. Particular emphasis is placed on the PhD students independently authoring project reports based on interpretation and evaluation of the numerical results obtained, with references and other documentation in support of the conclusions drawn.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to discretize ordinary and partial differential equations using finite difference and finite element methods, and to be able to independently implement and apply such algorithms
- be able to independently proceed from observation and interpretation of results to conclusion, and be able to present and account for his or her conclusions on a scientific basis in free report format.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to independently, on a scientific basis, select suitable computational algorithms for given problems
- be able to apply such computational algorithms to problems from applications
- be able to independently evaluate the relevance and accuracy of computational results
- be able to present solutions of problems and numerical results in written form.

*Judgement and Approach*

For a passing grade the doctoral student must

- be able to write a logically well structured report in suitable terminology on the construction of basic numerical methods and algorithms
- be able to independently evaluate obtained numerical results in relation to the (unknown) solution of the differential equation studied
- be able to independently author project reports of scientific character, with references and other documentation of work carried out in support of his or her conclusions.

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.

Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, 2009. ISBN 9780521734905.

**Types of instruction:** Lectures, seminars

**Examination formats:** Written exam, written assignments**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Calculus in one and several variables, linear algebra, basic theory for systems of linear differential and difference equations, basic theory for the partial differential equations of mathematical physics.

**Course coordinators:** **Web page:** http://www.ctr.maths.lu.se/course/NUMN12/