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Forum: Fields: Magnetic and Otherwise
Re: None Question about rough, smooth field! (LEE JaeHa)
Date: 14 Jul, 2015
From: John Apostolakis <John Apostolakis>

Dear JaeHa,

In this case smoothness means that a function is continuous and has several derivatives.  It particular that it has a continuous first derivative, and also a finite second derivative.

Under these conditions the Taylor expansion upon which the Runge-Kutta method depends will be well behaved, and high order methods can be relied upon. 

The opposite case is when the field is modelled with a discontinuous function, or one in which the first derivative is discontinuous.  ( This is not physical, but it could be an approximation of a field which is physical. )  In that case the Runge-Kutta methods of higher orders (4, 5 or more) would have difficulty coping at the location the discontinuity or discontinuities.  A method of lower order (2 or 3) will definitely cope better at those locations;  so on average it may perform better with the overall integration.

The number of integration steps and thus the performance will also significantly depend on the accuracy which you require.  If you require lower accuracy, which is relative error ~ 10 ^ -3  ( = 0.001 ) to 10 ^ -4 then the lower order methods will perform best.  If you require higher accuracy then the higher order methods will perform best ( G4CashKarpRKF45 is especially good, and G4ClassicalRK4 is good too.) 

In general I would recommend trying both lower order (3rd G4SimpleHeum or 2nd G4SimpleRunge) and higher order methods (G4ClassicalRK4 or G4CashKarpRKF45) if you have a hard problem (non-smooth fields) and seeing which method gives you the best overall performance.

Best regards,
John

> On 14 Jul 2015, at 09:06, LEE JaeHa <jaeha20@nuclearemail.org> wrote:
> 
> 
> *** Discussion title: Fields: Magnetic and Otherwise
> 
> in user guide, " if the field is calculated from a field map, a lower
> order stepper is recommended."
> 
> So i know that if the field is calculated from a field map, it's rough filed.
> 
> but i don't understand about definition of rough, smooth field.
> 
> could you explain for me? ..
> 
> -------------------------------------------------------------
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