Course unit
Last updated: 09/12/2024
EditDuring this introduction unit students will learn how to analyse and model a physical problem in order to obtain Partial Differential Equations (PDE) to describe it. From a mathematical point of view these equations are classified in three categories (elliptical, parabolic and hyperbolic). The most known physical representations are the stationary diffusion equation, the non-stationary diffusion equation and the wave equation. Even a succinct mathematical study will allow the characterisation of the solution behaviour of these PDE and link them to the physical behaviour we wish to model. In addition this allows us to anticipate the properties which the discretisation method must satisfy to achieve the best numerical solution.
| On completion of the unit, the student will be capable of: | Classification level | Priority |
|---|---|---|
| Establishing a balance equation | 4. Analyse | Essential |
| Recognizing a PDE type | 3. Apply | Essential |
| Knowing the properties a numerical scheme should demonstrate, depending on the type of PDE | 3. Apply | Essential |
| Percentage ratio of individual assessment | Percentage ratio of group assessment | ||||
|---|---|---|---|---|---|
| Written exam: | 100 | % | Project submission: | 0 | % |
| Individual oral exam: | 0 | % | Group presentation: | 0 | % |
| Individual presentation: | 0 | % | Group practical exercise: | 0 | % |
| Individual practical exercise: | 0 | % | Group report: | 0 | % |
| Individual report: | 0 | % | |||
| Other(s): 0 % | |||||
| Type of teaching activity | Content, sequencing and organisation |
|---|---|
| Course | From physical modelling to mathematical analysis (9h, Julien Bruchon) Objective: introduce, via. detailed descriptions of physical examples, the three types of second order partial differential equations, namely elliptical, parabolic and hyperbolic equations. The emphasis will be placed on aspects of the law of conservation, to obtain un-stationary convection diffusion equations, with for example problems of diffusion (particles or heat), the Fokker-Planck equation or the balance equation of a rubber membrane. The description of the propagation of sound will lead to the introduction of wave equations and hence hyperbolicity. The link with first order hyperbolic equations will be made. Following each example, a simplified mathematical analysis will allow us to specify the conditions under which the problem is set and to identify the principle properties of the solution (regularity, appearance of discontinuity, etc). Finally a more formal mathematical summary will lead to the introduction of notions of characteristic curves, to classify the operations and to express them in their standard form. |
| Course | Introduction to discretisation methods of PDE (E. Touboul, 6h) Objective : to introduce the general principles of PDE discretisation: Cauchy – Lipschitz theorem, approximation method: consistence, convergence and stability. Choice of discretisation methods Linear algebra (own values, matrix condition, Gaussian elimination, iterative methods) |