# FLASH4.6.2 API

Generated from /asc/asci2/site/flashcode/secure/release_4p6/source/numericalTools/Interpolate/Interpolate_cubic3DmonoDerv.F90 with ROBODoc v4.99.8 on Tue Oct 15 13:28:03 2019

## [Functions] source/numericalTools/Interpolate/Interpolate_cubic3DmonoDerv

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NAME

```  Interpolate_cubic3DmonoDerv
```

SYNOPSIS

```  call Interpolate_cubic3DmonoDerv (integer, intent (in)  :: nx,
integer, intent (in)  :: ny,
integer, intent (in)  :: nz,
real,    intent (in)  :: f    (-2:nx+3 , -2:ny+3 , -2:nz+3),
real,    intent (out) :: fx   (-1:nx+2 , -1:ny+2 , -1:nz+2),
real,    intent (out) :: fy   (-1:nx+2 , -1:ny+2 , -1:nz+2),
real,    intent (out) :: fz   (-1:nx+2 , -1:ny+2 , -1:nz+2),
real,    intent (out) :: fxy  ( 0:nx+1 ,  0:ny+1 ,  0:nz+1),
real,    intent (out) :: fxz  ( 0:nx+1 ,  0:ny+1 ,  0:nz+1),
real,    intent (out) :: fyz  ( 0:nx+1 ,  0:ny+1 ,  0:nz+1),
real,    intent (out) :: fxyz ( 1:nx   ,  1:ny   ,  1:nz  ))
```

DESCRIPTION

```  Calculates 1st order x,y,z derivatives, 2nd order xy,xz,yz mixed derivatives and
3rd order xyz mixed derivatives from a collection of cube data points, such that the
resulting coefficients for a triicubic expansion deliver a monotone 3D surface. While
there is always one such monotone surface if all derivatives are set equal to zero, the
resulting 3D surface is bumpy, especially on data points representing surfaces of constant
slope in either x,y or z direction. The present routine calculates derivatives, which
corrects for this bumpiness and delivers a more smooth monotonic surface. Since the code
is dealing with cubes in terms of rescaled [0,1] x,y,z coordinates, the resulting derivatives
will all be rescaled.

Labeling convention of all variables according to cube grid point location (only the
positive y-axis region is shown for clarity):

mpp -------- 0pp -------- ppp
/|           /|           /|
/ |          / |          / |
/  |         /  |         /  |
/   |        /   |        /   |
/    |       /    |       /    |        z (k index)
m0p -------- 00p -------- p0p    |
|    mp0 ----|--- 0p0 ----|--- pp0           |
|    /|      |    /|      |    /|            |  y (j index)
|   / |      |   / |      |   / |            |
|  /  |      |  /  |      |  /  |            |  /
| /   |      | /   |      | /   |            | /
|/    |      |/    |      |/    |            |/
m00 -------- 000 -------- p00    |             --------- x (i index)
|    mpm ----|--- 0pm ----|--- ppm
|    /       |    /       |    /
|   /        |   /        |   /
|  /         |  /         |  /
| /          | /          | /
|/           |/           |/
m0m -------- 00m -------- p0m

For evaluating the 3rd order mixed derivative at a point (i,j,k), the nearest neighbor
2nd order derivatives are needed -> we need an extra layer of 2nd order derivatives
beyond the intended 3D grid. Evaluation of the 2nd order derivatives needs nearest neighbor
1st order derivatives -> we need 2 extra layers of 1st order derivatives beyond the
intended 3D grid. Evaluation of the 1st order derivatives needs nearest neighbor
data point values -> we need 3 extra layers of data point values beyond the intended
3D grid.
```

ARGUMENTS

```  nx           : number of intended 3D grid points in x direction
ny           : number of intended 3D grid points in y direction
nz           : number of intended 3D grid points in z direction
f    (i,j,k) : data value for i,j,k-th grid point
fx   (i,j,k) : 1st order x derivative value for i,j,k-th grid point
fy   (i,j,k) : 1st order y derivative value for i,j,k-th grid point
fz   (i,j,k) : 1st order z derivative value for i,j,k-th grid point
fxy  (i,j,k) : 2nd order mixed xy derivative value for i,j,k-th grid point
fxz  (i,j,k) : 2nd order mixed xz derivative value for i,j,k-th grid point
fyz  (i,j,k) : 2nd order mixed yz derivative value for i,j,k-th grid point
fxyz (i,j,k) : 3rd order mixed xyz derivative value for i,j,k-th grid point
```

NOTES

```  ...
```