Numerical Methods for Differential Equations
Numeriska metoder för differentialekvationer

FMNN10F, 7.5 högskolepoäng

Gäller från och med: Autumn 2019
Beslutad av: Professor Thomas Johansson
Datum för fastställande: 2019-10-08

Allmänna uppgifter

Avdelning: Numerical Analysis
Kurstyp: Gemensam kurs, avancerad nivå och forskarnivå
Kursen ges även på avancerad nivå med kurskoder: FMNN10, NUMN12
Undervisningsspråk: English

Syfte

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.

Mål

Kunskap och förståelse

För godkänd kurs skall doktoranden

• 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.

Färdighet och förmåga

För godkänd kurs skall doktoranden

• 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.

Värderingsförmåga och förhållningssätt

För godkänd kurs skall doktoranden

• 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.

Kursinnehåll

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.

Kurslitteratur

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

Kursens undervisningsformer

Undervisningsformer: Föreläsningar, seminarier

Kursens examination

Examinationsformer: Skriftlig tentamen, inlämningsuppgifter
Betygsskala: Underkänd, godkänd
Examinator:

Antagningsuppgifter

Förutsatta förkunskaper: 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.

Kurstillfällesinformation

Kontaktinformation och övrigt

Kursansvariga:
Hemsida: http://www.ctr.maths.lu.se/course/NUMN12/