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Finite Difference

Presentation

LINEAR PDE PART (analysis, numerics)

 

PDE models : examples ; the 4 fundamental linear PDE. Classification.

Principleof the Finite Difference method: consistancy, stability, convergence.

Laplace-Poisson equation (elliptic): explicit solution (by separation of variables), maximum principle, FD schemes.

Heat equation (parabolic): explicit solution (by Fourier transform), FD schemes  (explicit, implicit, splitting). A non linear case.

Transport equation (hyperbolic) : explicit solution -  characteristics, schemes, equivalent equation.

Waves equation (hyperbolic) : explicit solution - characteristics, schemes.

Practical works: stability, accuracy ; modeling.

 

MONTE-CARLO PART

-          Generation of random numbers, simulation by inversion of the distribution function, by the reject method and by some specific methods, Monte-Carlo Methods (convergence, rate of convergence, variance reduction by using different methods).

Objectives



- Be able to use standard analytical tools to study partial differential
equations (PDE), for direct and variational analysis of problems.
- Classify first- and second order-partial differential equations, and identify the
possible behaviours of solutions for the transport-diffusion problem.
- Use the notions of convergence, consistency and stability, whatever the kind of
discretization.
- Analyse convergence of finite difference schemes for time-dependent
problems, and be able to build a stability analysis in Fourier space by means of
symbol theory.
Apply these approximation methods to common first- and second-order partial differential
equations, and perform their implementation.

Needed prerequisite

All courses of 3A MIC, especially the ones focusing of numerical analysis, matrix analysis and ordinary differential equations (EDO).Course on mathematical Analysis (GMM4, semester 7).

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...