B6.2 Numerical Solution of Differential Equations II - Material for the year 2019-2020

2019-2020
General Prerequisites: 

Part A Differential Equations 1. B5.2 Applied Partial Differential Equations is desirable but not essential.

Course Term: 
Hilary
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
H

Assessment type:

Course Overview: 

To introduce and give an understanding of numerical methods for the solution of hyperbolic and elliptic partial differential equations, including their derivation, analysis and applicability.

Learning Outcomes: 

At the end of the course the student will be able to:

  1. construct practical methods for the numerical solution of boundary-value problems arising from ordinary differential equations and elliptic partial differential equations; analysis of the stability, accuracy, and uniqueness properties of these methods,
  2. construct methods for the numerical solution of initial-boundary-value problems for first- and second-order hyperbolic partial differential equations, and to analyse their stability and accuracy properties.
Course Synopsis: 

The course is devoted to the development and analysis of numerical solutions of boundary value problems for second-order ordinary differential equations, boundary-value problems for second-order elliptic partial differential equations, and initial-boundary-value problems for first- and second-order hyperbolic partial differential equations. The course begins by considering classical techniques for the numerical solution of boundary-value problems for second-order ordinary differential equations and elliptic boundary-value equations, in particular the Poisson equation in two dimensions. Topics include: discretisations (e.g., finite difference, finite element, and spectral methods), error and convergence analysis, and the use of maximum principles. The remaining lectures focus on the numerical solution of initial-boundary-value problems for hyperbolic partial differential equations with topics such as: discretisations (e.g., finite difference and finite volume), method of lines, accuracy, stability (including CFL condition) and convergence, limiters, total variation, WENO schemes and energy methods.

Reading List: 

The course will be based on the following textbooks:

  1. A. Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, second edition, 2009), Chapters 8-10, 17.
  2. R. LeVeque, Finite difference methods for ordinary and partial differential equations (SIAM, 2007). ISBN 978-0-898716-29-0 [Chapter 10].
  3. R. LeVeque, Numerical methods for conservation laws (Birkhaüser 1992), ISBN 0-8176-2723-5 [Chapters 10-16].