MAR572 ## Numerical Methods in Atmospheric Sciences

Unit 1. Fundamentals of Difference Schemes

Lecture 1: Basic concepts1. Definitions of consistence, convergence, and stability

2. The Lax Equivalence Theorem

Lecture 2: Design of difference schemes1. 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 analysis1. Matrix method

2. Energy method

3. The Von Neumann method

Lecture 4: Examples1. Eular scheme

Lecture 5: Some basic schemes for and their generalizations

2. Upstream and downstream schemes

3. Implicit schemes

4. Matsuno scheme

5. Lax Wendroff scheme

6. Forward-backward scheme

Lecture 6: Time splitting method1. The problem

2. The splitting method

3. Example

Lecture 7: Other computational problems1. Numerical solution of the advection equation

2. Discussions of numerical dispersion

Lecture 8: Arrangement of grid points1. 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 instability1. Fourier representation of discrete fields

2. Nonlinear interaction and instability

3. Methods to eliminate nonlinear instability

Lecture 10: Construction of conservation scheme1. A model

2. Various difference forms

3. Notes

Lecture 11: The barotropic vorticity model1. The equation

2. conservation properties

3. The Arakawa Jacobian

4. Time difference

Lecture 12: Spectral method1. Basis functions

2. A spectral model

3. Calculation procedure

4. Discussions about the method

Lecture 13: Spectral-transform method1. The principle

2. Example

3. About the boundary condition

Lecture 14: Spectral method on the sphere1. Spherical harmonics

2. Grid-spectral transformation

3. Gaussian quadrature formula

4. Truncation

Unit. 4. Methods to Solve Elliptic Equations

Lecture 15: Examples1. 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 method1. 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 fitting1. 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 method1. Review

2. Construction of variational forms

Lecture 19: Statistical Estimation1. Maximum likelihood estimation

2. Least variance estimation

Lecture 20: Statistical interpolation1. simple cases

2. general form

Lecture 21: Adjoint models I1. 1-dimensional case

2. 2-dimensional linear difference models

Lecture 22: Adjoint models II1. Linear differential equations

2. Non-linear differential equations

3. Applications

Lecture 23: Initialization of numerical models I1. 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