Resource pages for GEOF211 - Numerical Modeling

This is a compendium for use in in the course GEOF211 at the Geophysical Institute, University of Bergen.

In this course you learn to use numerical methods to solve simple problems in fluid dynamics. We study exact and numerical solutions of partial differential equations linked to advection, diffusion, oscillation and wave processes, and discuss the strengths and weaknesses of different numerical schemes.

You will learn to use methods to solve initial value problems and solve diagnostic boundary value problems.

The compendium is a collection of notes by Kjersti Daae, Knut Barthel, João Bettencourt, and Thorbjørn Østenby Moe

You can cite this work by writing, e.g., Daae, Bettencourt, and Moe [DBM25], chapter 1.2 in your text, and add the following to the reference list at the end of the assignment:

Kjersti Daae, João Bettencourt, and Thorbjorn Ostenbye Moe. Resource pages for GEOF211 - numerical modeling. webpage address, Month Year.

• 1. Useful calculus
  • 1.1. Taylor series
  • 1.2. Trigonometry
  • 1.3. Absolute values
  • 1.4. Complex numbers
  • 1.5. The Euler formula
  • 1.6. Fourier transforms
  • 1.7. Bonus example code for visualisation of the Fourier components!
  • 1.8. The Formal definition of the derivative
  • 1.9. Differential equations
• 2. Geophysical Fluid Dynamics (GFD)
  • 2.1. The Primitive equations
  • 2.2. The Advection equation
  • 2.3. The diffusion equation
  • 2.4. Inertial oscillations
  • 2.5. The Shallow water equations
  • 2.6. The wave equation
• 3. The finite difference method (FDM)
  • 3.1. Algebraic approximation of the first derivative
  • 3.2. Truncation Error
  • 3.3. Estimates of the truncation error
  • 3.4. Finite difference formulas for the 1st derivative
  • 3.5. Finite difference formulas for the 2nd derivative
• 4. Stability of numerical schemes
  • 4.1. Concepts and definitions
  • 4.2. Domain of Dependence (DoD)
  • 4.3. Courant-Friedrich-Lewy, CFL criterion
  • 4.4. Total Variation (TV) and Total Variation Diminishing (TVD) schemes (part 1)
  • 4.5. Von Neumann Stability Analysis
• 5. Grids
  • 5.1. Staggered grids
• 6. Implicit schemes vs explicit schemes
  • 6.1. The Crank-Nicholson Scheme for linear advection
  • 6.2. The Implicit Scheme
• 7. Boundary conditions
  • 7.1. Mathematical descriptions of boundary conditions
  • 7.2. Physical descriptions of boundary conditions
• 8. The advection equation
  • 8.1. The exact equation
  • 8.2. The difference equation
  • 8.3. Solution of the difference equation
  • 8.4. Analysis of the solution of the difference equation
  • 8.5. Schemes with stabilizing terms
  • 8.6. Predictor-corrector schemes
  • 8.7. Half-step schemes
  • 8.8. The Lax-Wendroff 1-step approach from Taylor expansions
• 9. Total Variation Diminishing schemes
  • 9.1. Godunov’s method
  • 9.2. Flux Limiters
  • 9.3. Godunov schemes with TVD flux limiters
• 10. Diffusion
  • 10.1. The diffusion equation
  • 10.2. The Crank-Nicholson Scheme for diffusion
  • 10.3. von Neumann stability analysis for the diffusion equation
• 11. Waves
  • 11.1. The classical wave equation and its solution
  • 11.2. von Neumann analysis of the wave equation
• 12. References