module NumRu::Derivative in derivative.rb

todo

• decide argument of b_expand_linear_ext is Symbol or Numeric.
  • now is Numeric.
  • it denpends the treatment of dRuby.
• support other boundary conditions.

Index

• module NumRu::Derivative
  • threepoint_O2nd_deriv
    • First derivative (2nd Order difference use three point.)
  • cderiv
    • First derivative (center difference use two point.)
  • deriv2d
    • 2nd derivative
  • b_expand_linear_ext
    • return array extended boundaries with linear extention.
  • cdiff
    • return difference. (center difference)

module NumRu::Derivative

Module functions of Derivative Operater for NArray.

threepoint_O2nd_deriv(z, x, dim, bc=LINEAR_EXT)

Derivate of z with respect to dim th dim using a 2nd order 3-point differentiation valid for non-uniform grid:

(s**2*z_{i+1} + (t**2 - s**2)*f_{i} - t**2*f_{i-1}) / (s*t*(s + t))

Here, s represents (x_{i} - x_{i-1}) ,t represents (x_{i+1} - x_{i}) and _{i} represents the suffix of {i} th element in the dim th dimension of the array. ).

ARGUMENTS

• z (NArray): a NArray which you want to derivative.
• x (NArray): a NArray represents the dimension which derivative respect to. z.rank must be 1.
• dim (Numeric): a Numeric represents the dimention which derivative respect to. you can give number count backward (dim<0), but z.rank ��dim must be > 0.
• bc (Numeric) : a Numeric to represent boundary condition. Supported conditions are expressed in terms of boundary extension applied before differentiation:
  • LINEAR_EXT(=1): default value. linear extension using two grid points at the boundaries.
  • CYCLIC_EXT(=2): cyclic extension as k,0,1,..,k,0 for indices 0,1,..,k
  • MIRROR_A(=3): for indices 0,1,..,k, extends as 0,0,1,..,k,k (good for cell interior values with the Neumann condition)
  • MIRROR_B(=4): for indices 0,1,..,k, extends as 1,0,1,..,k,k-1 (good for cell boundary values with the Neumann condition)

RETURN VALUE

• O2nd_deriv_data (NArray): (s**2*z_{i+1} + (t**2 - s**2)*f_{i} - t**2*f_{i-1}) / (s*t*(s + t))

cderiv(z, x, dim, bc=LINEAR_EXT)

Derivate of z with respect to dim th dim using centeral differenciation: (z_{i+1} - z_{i-1}) / (x_{i+1} - x_{i-1})

ARGUMENTS

• z (NArray): a NArray which you want to derivative.
• x (NArray): a NArray represents the dimension which derivative respect to. z.rank must be 1.
• dim (Numeric): a Numeric represents the dimention which derivative respect to. you can give number count backward (dim<0), but z.rank ��dim must be > 0.
• bc (Numeric) : a Numeric to represent boundary condition. See threepoint_O2nd_deriv for supported conditions.

RETURN VALUE

• cderiv_data (NArray): (z_{i+1} - z_{i-1}) / (x_{i+1} - x_{i-1})

deriv2nd(z, x, dim, bc=LINEAR_EXT)

2nd Derivate of z with respect to dim-th dim covering non-uniform grids. Based on:

( (z_{i+1}-z_{i})/(x_{i+1}-x_{i}) - (z_{i}-z_{i-1})/(x_{i}-x_{i-1}) )
/ ((x_{i+1}-x_{i-1})/2)

ARGUMENTS

• z (NArray): a NArray which you want to derivative.
• x (NArray): a NArray represents the dimension which derivative respect to. z.rank must be 1.
• dim (Numeric): a Numeric represents the dimention which derivative respect to. you can give number count backward (dim<0), but z.rank ��dim must be > 0.
• bc (Numeric) : a Numeric to represent boundary condition. See threepoint_O2nd_deriv for supported conditions.

RETURN VALUE

• cderiv_data (NArray): (z_{i+1} - z_{i-1}) / (x_{i+1} - x_{i-1})

b_expand_linear_ext(z, dim)

expand boundary with linear value. extend array with 1 grid at each boundary with dim th dimension, and assign th value which diffrential value between a grid short of boundary and boundary grid in original array. (on other wards, 2*z_{0}-z_{1} or 2*z_{n-1}-z_{n-2}: now _{i} represents the

suffix of {i} th element in the ((<dim>)) th dimension of array. ).

ARGUMENTS

• z (NArray): a NArray which you want to expand boundary.
• dim (Numeric): a Numeric represents the dimention which derivative respect to. you can give number count backward (dim<0), but z.rank ��dim must be > 0.

RETURN VALUE

• expand_data (NArray):

cdiff(x, dim)

Diffrence operater. return an NArray which a difference x ( in other wards, (x_{i+1} - x_{i-1}): now _{i} represents the suffix of {i} th element in the dim th dimension of array. ).

ARGUMENTS

• x (NArray): a NArray which you want to get difference.
• dim (Numeric): a Numeric representing the dimention which derivative respect to. you can give number count backward (dim<0), but z.rank ��dim must be > 0.

RETURN VALUE

• cdiff_data (NArray): (x_{i+1} - x_{i-1})