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# Numerical Methods in Earth Sciences

ECTS: 3

Unit administrator : F. Dubuffet

TeachersF. DubuffetV. LangloisA. Morison

Class type : 30hrs of courses and practical classes

Language: english or french

Contents:

Lectures : 14h

• Partial differential equations
• Finite-difference method
1. Taylor series - Accuracy and order
2. Stability analysis - Explicit and implicit Euler scheme, Crank-Nicolson scheme and Alternating direction implicit schemes
3. Modified equation method: numerical dissipation and dispersion
4. Applications to parabolic and hyperbolic equations

Fig: Numerical solution of the 1-D advection equation using the Beam Warming scheme. Analytical solution (dotted line) and numerical solution for 2 values of the Courant number (C=0.5 in bold and C=1.5)

• Finite Volume

Fig: example of a staggered grid in 2-D cartesian

1. Introduction
2. Applications to the advection/diffusion equation
3. Irregular grids
4. Example: MPDATA scheme

• Spectral and pseudo-spectral methods
1. Introduction
2. Fourier transform
3. Order and limitation

Fig: Temperature field in a model of Rayleigh-Bénard convection with an infinite Prandtl number, in the Boussinesq approximation and a Rayleigh number of 107. An alternating directiion implicit scheme and finite volumes are used to solve the heat equation. The stockes equation is solved using pseudo-spectral methods.

Practicals: 16h

The students write a Fortran 90 or a C program to solve a geophysical problem using some previous numerical methods.

### Ressources*:

• lectures (F. Dubuffet) [pdf]
• MPDATA scheme (P. K. Smolarkiewicz) [pdf]
• Summary (11/21/18 update) [pdf]
• Fortran 90 [pdf]
###### *Connection obligatoire pour récupérer les documents

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Département ENS
• Directeur :
Stéphane Labrosse
• Secrétaire :
Emmanuelle Lousson
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