Partial differential equations and numerical methods [SMATB317]

Learning outcomes

The course will provide students with a theoretical and practical grounding in the treatment of partial differential equations: position of the problem, boundary conditions, classification, analytical solutions, numerical methods.

Objectives

By the end of the course, students should be able to:

- analyse a problem formulated in terms of PDEs - partial differential equations - (analytical properties and classification),

- find general or specific analytical solutions, possibly in simplified cases

- develop and validate a numerical code for approximating the solutions of PDEs.

Content

The theoretical course is divided into two main parts. The first deals with the theoretical aspects (position of a PDE problem, classification, analytical methods); the second presents two large families of basic methods for their numerical 'solution' (approximation), with solved examples. The practical work will cover both the analytical aspects ('hand solving') and the numerical aspects ('computer solving').

Table of contents

Part 1: Theoretical aspects of partial differential equations (PDEs) - C. Dubussy

    Notions of PDEs and well-posed problems (Dirichlet and Neumann conditions).
    Examples of easy solutions of PDEs of order 1.
    Characterisation of PDEs of order 2.
    The homogeneous wave equation, first without initial values, then with (d'Alembert).
    The homogeneous diffusion equation, the maximum principle, the case of the line, the half-line, etc.
    Inhomogeneous equations, method of characteristics, Duhamel's principle.
    The case of segments: the method of separation of variables.
    Laplace equation. Study in dimension 2 on a rectangle, a circle, etc.

Part 2: Introduction to numerical methods for solving PDEs (A. Füzfa)

    Finite difference method: construction of approximation formulae; truncation and rounding errors by example.
    Edge conditions (Dirichlet, Neumann, Sommerfeld-radiative) and implementation
    Application of finite differences: line method, finite difference schemes, illustrations on first- and second-order PDEs, convergence and validation of codes (analytical solutions, conserved quantities, etc.), Courant-Friedrich stability conditions
    Spectral methods; principles; function approximations by orthogonal decomposition in a Hilbert space; eigenfunctions of the Laplacian in different geometries and coordinates; application to the wave or heat equation