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

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

16 lectures

### Assessment type:

- Written Examination

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.

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

- 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,
- 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.

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.

The course will be based on the following textbooks:

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