Finite-difference Errors - Part 1: Amplitude Error Analysis of Linear Eqs; the von Neuman method

Under construction (this web page, and most other web pages for this course).

Instructor: Roland Stull

Learning Goals:   By the end of this module, you will be able to ...

• Explain the meanings of ...
• truncation error,
• linear amplitude error,
• linear phase error,
• nonlinear error.
• define the Courant number and explain why it is relevant to NWP.
• use von Neumann stability analysis on any finite difference scheme to determine linear stability.
• compare errors for advection vs. diffusion terms in the weather forecast eqs.
  
  

Readings BEFORE class:

1. Stull p 759-761, and Warner p72-118.
2. Press et al "Numerical Recipes 3rd Ed", Chapter 20 PDEs, start on p1032. List of ...

Topics

A. List of Error Types.

B. Truncation Error.  

1. Centered difference as an example

C. Amplitude Errors for Linear eqs.

1. von Neumann stability analysis method
   
2. The linear advection eq.
3. Stability analysis of different explicit  finite-diff approximations (from the list below) for the linear advection eq.
4. Diffusion eq. -- stability analysis of different finite-diff approximations.
   a) forward in time;
   b) centered in time. approximations.
5. Implicit finite diff  schemes-- stability analyses:
   a) Advection example using Crank-Nicholson method.
   b) Diffusion example.
   c) Diffusion using Crank-Nicholson

    List of some of the Explicit Finite Difference Approximations

1. Euler-forward in time, centered in space.
2. Forward in time, backward in space.
3. Forward in time from spatial average (Lax method), centered in space.
4. Centered in time, 3-point centered in space (Leapfrog method)
5. Centered in time, 5-point centered in space
   
6. Two-step Lax-Wendroff
7. Lax single-step in 2-D
8. 3rd-order Runge-Kutta in time, 2nd-order centered in space (WRF-ARW)
9. Adams-Bashforth (2nd order in time)
10. Off-centered Adams-Bashforth (WRF-NMM)