Chombo + EB + MF  3.2
Classes | Macros | Functions
FourthOrderMappedFineInterpSup.H File Reference

Supporting classes and routines for FourthOrderMappedFineInterp. More...

#include <cstring>
#include "NamespaceHeader.H"
#include "NamespaceFooter.H"
Include dependency graph for FourthOrderMappedFineInterpSup.H:

Go to the source code of this file.


class  CoordTransform
 Coordinate transformations. More...




int binomial (const int n, int k)
 Calculate a binomial coefficient. More...
int powerIndex (int a_m, IntVect a_p)
 Find the sequential index of a power. More...

Detailed Description

Supporting classes and routines for FourthOrderMappedFineInterp.

Macro Definition Documentation


Function Documentation

int binomial ( const int  n,
int  k 

Calculate a binomial coefficient.

Binomial coefficient

References CH_assert.

Referenced by powerIndex().

int powerIndex ( int  a_m,
IntVect  a_p 

Find the sequential index of a power.

The powers are indexed as follows (i.e for $d=3$ dimensions)

*     int idx = 0
*     for (px = 0; px <= m; ++px)
*       for (py = 0; px+py <= m; ++py)
*         for (pz = 0; px+py+pz <= m; ++py)
*           ++idx                                                  

where $m$ is the degree of the polynomial and we wish to find idx. To compute the sequential index of any given power, we can use the relations

\[ \mbox{number of powers} = {m + d \choose d} \]


\[ \sum_{j=k}^n {j\choose k} = { n+1 \choose k+1 } \]

With these, the amount to add to the sequential index for a power at a spatial index is total number of powers remaining at this spatial index (remainder of $m$) minus the number of powers not used at this spatial index. E.g, if $m=3$, $d=3$, and $p_x=1$, there are 2 powers left for the remaining 2 dimensions,

\[ {2+2 \choose 2}\,. \]

The increment to the sequential index is then

\[ {2+3+1 \choose 2+1} - {2+2+1 \choose 2+1} \]

In general, this can be written for direction index $i$, in $d$ space dimensions, with $m_i$ giving the remaining available powers at $i$, and $p_i$ giving the power used at index $i$ as

\[ {d-i+m_i \choose d-i} - {d-i+m_i-p_i \choose d-i } \,. \]

[in]a_mDegree of the polynomial
[in]a_pPower for each direction
Sequential index

References binomial(), and SpaceDim.