English

Every autumn semester

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.

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.

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.

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.

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.

Lectures

Seminars

Written exam

Written assignments

Failed, pass

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.

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

FMNN10F

2019-10-08

Professor Thomas Johansson