Finite-difference Errors - Part 2: Amplitude Error Analysis of Linear Eqs; the von Neuman method (continued_

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 ...

• use stability analyses to recommend which finite-different schemes are best.
• discuss the advantages and disadvantages of the finite difference schemes used in the WRF model.
• compare the finite difference schemes used in WRF-ARW vs. WRF-NMM.
• identify the appropriate stability criteria for the diffusion eq.
• compare explicit vs. implicit time-differencing schemes.
  
  

Readings BEFORE class (same as previous readings):

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 ...

Homework AFTER class:   Homework 5 

1. Do a von Neumann linear stability analysis using the eigenmode method as we used in class (Press et al, Numerical Recipes), to answer the following questions regarding the 3-point-centered-in-time 5-point-centered-in-space approximation to the linear advection eq.:
   [T(j,n+1) - T(j,n-1)] / (2 ∆t) = - [Uo / (12∆x)] [T(j-2,n) - 8T(j-1,n) + 8T(j+1,n) - T(j+2,n)]
   

1. What criterion is required to ensure numerical stability.  (Hint, the answer of C_R ≤ 0.729 is in Warner p69 top line.)  But you need to show all the steps to get this answer. 
   
2. What is the amount of damping (amplitude reduction) as a function of wavenumber k (or as a function of m, for m∆x wavelengths)?

2. For the linear advection equation that is numerically approximated by a 3rd-order Runge-Kutta (RK3)-in-time with 2nd-order-centered-in-space scheme discussed in class:
1. Check the algebra of my in-class derivation of the single-equation version of RK3.
2. Do a von Neuman linear stability analysis of this single RK3 equation using the eigenmode method of Press et al (Numerical Recipes), and find the equation for the magnitude of the amplitude |A(k)| , or for  |A(k)|²  if easier.
3. For the worst case wavelength, what is the critical value of C_R (Courant number) is needed to maintain stability, and  what wavelength is the worst-case wavelength?
   
4. How does the Courant stability criterion for RK3 with 2nd-order-Spatial Differencing (that you just derived) compare with the higher-order stability criteria given in Table 3.1 (p28 of the WRF-ARW4.3 tech note; see link to it in the Resources web page)?
   
5. Plot  |A(k)|  vs. C_R (Courant number in the range of 0 to 3) for various wavelengths:  L =  2∆x,  2.5∆x,  3∆x,  4∆x,  5∆x,  10∆x,  20∆x.

Topics (same as last week)

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)