# B6.1 Numerical Solution of Differential Equations I - Material for the year 2020-2021

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A7: Numerical Analysis is desirable but not essential.

16 lectures

### Assessment type:

- Written Examination

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

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

- construct one-step and linear multistep methods for the numerical solution of initial-value problems for ordinary differential equations and systems of such equations, and to analyse their stability, accuracy, and preserved geometric properties;
- construct numerical methods for the numerical solution of initial-boundary-value problems for parabolic partial differential equations, and to analyse their stability and accuracy properties.

The course is devoted to the development and analysis of methods for numerical solution of initial value problems for ordinary differential equations and initial-boundary-value problems for second-order parabolic partial differential equations. The course begins by considering classical techniques for the numerical solution of initial value ordinary differential equations. The problem of stiffness is discussed in tandem with the associated questions of step-size control and adaptivity. Topics include: Euler, multistep, and Runge-Kutta methods; stability; stiffness; error control; symplectic and adaptive algorithms.

The remaining lectures focus on the numerical solution of initial-boundary-value problems for parabolic partial differential equations. Topics include: explicit and implicit methods; accuracy, stability and convergence, use of Fourier methods for analysis.

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). ISBN 978-0-521-73490-5 [Chapters 1-6, 16]. - R. LeVeque,
*Finite difference methods for ordinary and partial differential equations*(SIAM, 2007). ISBN 978-0-898716-29-0 [Chapters 5-9]. - E. Süli and D. Mayers,
*An Introduction to Numerical Analysis*(Cambridge University Press, 2006). ISBN 0-521-00794-1 [Chapter 12].