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## Correcting for Staggered Leapfrog

In practice the program uses a staggered leapfrog algorithm so in solving for the value of is known at where is the time step. See section 6.4 for more details. The solution to this problem is to use the two equations

 (6.27)

where and refer to the values of at and respectively and all other variables are evaluated at time . Take the evolution equation to be
 (6.28)

Plugging this form into equation (6.27) and eliminating gives
 (6.29)

 (6.30) (6.31)

To determine whether to use the plus or minus sign in equation (6.31) consider the limit as . In this limit
 (6.32)

This suggests that the plus sign must be used in order to reduce to the limit . Hence
 (6.33)

In the program it's useful to calculate , which is roughly , so
 (6.34)

Thus equation (6.26) becomes
 (6.35)

where
 (6.36)

Next: Power-Law Expansion Up: Scale Factor Evolution Previous: The Scale Factor Equation

Send email to Gary Felder at gfelder@email.smith.edu
Send email to Igor Tkachev at Igor.Tkachev@cern.ch

This documentation was generated on 2008-01-21