# 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...