MAR572

Numerical Methods in Atmospheric Sciences

Unit 1. Fundamentals of Difference Schemes
Lecture 1: Basic concepts
1. Definitions of consistence, convergence, and stability
2. The Lax Equivalence Theorem
Lecture 2: Design of difference schemes
1. First and second order derivatives
2. Construction of higher order approximations of
3. higher order derivatives
4. Notes about dx and dt


Unit. 2. Methods for Initial-Value Problems of Linear Partial Differential Equations
Lecture 3: Stability analysis
1. Matrix method
2. Energy method
3. The Von Neumann method
Lecture 4: Examples
Lecture 5: Some basic schemes for and their generalizations
1. Eular scheme
2. Upstream and downstream schemes
3. Implicit schemes
4. Matsuno scheme
5. Lax Wendroff scheme
6. Forward-backward scheme
Lecture 6: Time splitting method
1. The problem
2. The splitting method
3. Example
Lecture 7: Other computational problems
1. Numerical solution of the advection equation
2. Discussions of numerical dispersion
Lecture 8: Arrangement of grid points
1. A 1-D model
2. Dispersion in grid system (A)
3. Other grid systems
4. 2-D cases


Unit 3. Methods for Nonlinear Initial-Value Problems
Lecture 9: Nonlinear computational instability
1. Fourier representation of discrete fields
2. Nonlinear interaction and instability
3. Methods to eliminate nonlinear instability
Lecture 10: Construction of conservation scheme
1. A model
2. Various difference forms
3. Notes
Lecture 11: The barotropic vorticity model
1. The equation
2. conservation properties
3. The Arakawa Jacobian
4. Time difference
Lecture 12: Spectral method
1. Basis functions
2. A spectral model
3. Calculation procedure
4. Discussions about the method
Lecture 13: Spectral-transform method
1. The principle
2. Example
3. About the boundary condition
Lecture 14: Spectral method on the sphere
1. Spherical harmonics
2. Grid-spectral transformation
3. Gaussian quadrature formula
4. Truncation


Unit. 4. Methods to Solve Elliptic Equations
Lecture 15: Examples
1. The stream function
2. Steady solution of a forced system
Relaxation method
1. Simultaneous relaxation
2. Sequential relaxation
3. SOR method
Lecture 16: Direct factorization method
1. Tri-diagonal matrix
2. Block tri-diagonal matrix
3. Reduction of 3-D elliptic equations to 2-D equations.


Unit. 5. Data Analysis
Lecture 17: Least square fitting
1. least square fitting
2. Straight-line data with errors in both coordinates
3. Confidence limits on estimated model parameters
4. Nonlinear models
Lecture 18: The variational method
1. Review
2. Construction of variational forms
Lecture 19: Statistical Estimation
1. Maximum likelihood estimation
2. Least variance estimation
Lecture 20: Statistical interpolation
1. simple cases
2. general form
Lecture 21: Adjoint models I
1. 1-dimensional case
2. 2-dimensional linear difference models
Lecture 22: Adjoint models II
1. Linear differential equations
2. Non-linear differential equations
3. Applications
Lecture 23: Initialization of numerical models I
1. Introduction
2. The linearized shallow water model
3. Normal modes
4. Dynamic initialization


Unit. 5. Summary
Lecture 24: Introduction to numerical design of general circulation models
Lecture 25: Review