OCG 593 Numerical Methods for Ocean and Atmospheric Modeling

Tentative Syllabus

Fall, 2009

I         Introduction

    1. Philosophy and words of inspiration (and caution).
    2. The modeler as a frustrated artist
    3. Course description: build an understanding of basics.
    4. Objective is to be able to apply numerical techniques to research problems.

II        Planning a numerical model

    1. Equations of motion and heat transfer in the ocean and atmosphere
    2. Motion and heat equations as sums of parabolic, advective, and elliptic equations.

III       Basic elements of the finite difference method

    1. Taylor series expansions
    2. The truncation error of a finite difference scheme
    3. Basic ideas of consistency, convergence and stability

IV       Parabolic heat equation as a case study

    1. Analytical solution for simple cases
      1. Explicit scheme (truncation error, convergence, stability)
      2. Implicit method (truncation error, convergence, stability)
      3. Thomas algorithm
      4. Weighted averaged method
      5. Three-time-level schemes
    1. More general linear problems
    2. Parabolic equations in two dimensions
      1. Alternating Direction Implicit methods

V        Advection equation

    1. Analytical solution for simple cases
    2. Time differencing schemes (Euler, leapfrog, Adams-Bashforth, Matsuno, Heun)
    3. Stability analysis, CFL condition
    4. Spatial approximations (upstream, downstream)
    5. Trapezoidal scheme, Lax-Wendroff scheme
    6. Analysis of the phase and group velocity changes in the numerical solution
    7. Two-dimensional advection equation

VI        Numerical methods for solving equations describing gravity and inertia-gravity waves

    1. One-dimensional problem (dispersion relation, computational dispersion)
    2. Two-dimensional problem (A, B, and C spatial grid configurations)
    3. Time integration schemes (explicit, implicit, forward-backward, leapfrog, Crank-Nicolson)
    4. The splitting Marchuk method

VII      Numerical solution of the barotropic vorticity equation

    1. Conservation laws
    2. Finite difference approximation of the Jacobian operator
    3. Conservative schemes and "box" method

VIII.    Elliptic equations

    1. Finite difference approximation
    2. Jacobian iteration. Gauss-Seidel iteration. (Convergence, stability)
    3. Successive over-relaxation (SOR) and Alternating direction implicit (ADI) methods

IX       Spectral (Galerkin) Methods

    1. Basic Chebyshev spectral techniques (discretization schemes, basic functions)
    2. Projections (Galerkin, Tao, collocation techniques)
    3. When to use spectral methods. Application to the linear advection equation

X.       Finite elements methods

    1. Application to the barotropic vorticity equation
    2. Application to the advection equation
    3. Comparison between spectral and finite element methods

XI.      General Circulation Models

    1. Ocean GCM models
    2. Atmospheric GCM models
    3. Coupled ocean-atmosphere models
    4. Data assimilation problems.

Grading: There will be two modeling problems (two-week assignment each, written report), a midterm exam and a final. They will count toward the grade as follows.

Modeling problem #1     20%

Modeling problem #2     20%

Midterm                        25%

Final                             35%