Numerical Solution of Differential Equations II

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

The MT lectures are devoted to numerical methods for initial value problems, while the HT lectures concentrate on the numerical solution of boundary value problems.

Students will understand and have experience of the theory for:

• Construction of shooting methods for boundary value problems in one independent variable
• Elementary numerical analysis of elliptic partial differential equations
• Analysis of iterative methods for solution of large linear systems of equations

The HT part of the course is concerned with numerical methods for boundary value problems. We begin by developing numerical techniques for the approximation of boundary value problems for second-order ordinary differential equations.

Boundary value problems for ordinary differential equations: shooting and finite difference methods. [Introduction (1 lecture) + 2 lectures]

Then we consider finite difference schemes for elliptic boundary value problems. This is followed by an introduction into the theory of direct and iterative algorithms for the solution of large systems of linear algebraic equations which arise from the discretisation of elliptic boundary value problems.

Boundary value problems for PDEs: finite difference discretisation; Poisson equation. Associated methods of sparse numerical algebra: sparse Gaussian elimination, classical iterations, multigrid iterations. [13 lectures]

Boundary value problems for ordinary differential equations: shooting and finite difference methods.

Boundary value problems for PDEs: finite difference discretisation; Poisson equation. Associated methods of sparse numerical algebra: brief consideration of sparse Gaussian elimination, classical iterations, multigrid iterations.

This course does not follow any particular textbook, but the following essentially cover the material:

• A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge U. Press, 1996, Chapters 7,10,11.
• K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994. Or the more recent 2nd edition (2005), Chapters 6 and 7.

Also

• G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press, Oxford, 1985 (and any later editions) has some of the material in chapter 5.