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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

 

Beam-Warming

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

Maillage décalé sur grille régulière

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*:

  • lecture 11/15/19 [pdf]
  • Smolarkiewicz, P.K. and L.G. Margolin, MPDATA: A Finite-Difference Solver for Geophysical Flows, J. Comp. Phys., 140, 459-480 (1998) [pdf]
  • Fortran [pdf]
  • Navier-Stokes practicals [pdf]
*Connection obligatoire pour récupérer les documents

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