get_numerical_derivative Subroutine

public subroutine get_numerical_derivative(x, y, dy, dxtol)

Calculates the numerical derivative of a given array. A sixth-order accurate central difference stencil is used to calculate the derivative. Near the edges a sixth-order accurate forward and backward difference stencil is used for the left and right boundary, respectively. It is assumed that the x values are all equally spaced. If this is not the case, a polynomial interpolation on a uniform grid can be done and that one can be differentiated instead. The stencils are as follows:

• 6th order central differences: $$dy_i = \frac{-y_{i-3} + 9y_{i-2} - 45y_{i-1} + 45y_{i+1} - 9y_{i+2} + y_{i+3}}{60dx}$$
• 6th order forward differences: $$dy_i = \frac{-147y_i + 360y_{i+1} - 450y_{i+2} + 400y_{i+3} - 225y_{i+4} + 72y_{i+5} - 10y_{i+6}}{60dx}$$
• 6th order backward differences: $$dy_i = \frac{10y_{i-6} - 72y_{i-5} + 225y_{i-4} - 400y_{i-3} + 450y_{i-2} - 360y_{i-1} + 147y_i}{60dx}$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp),	intent(in)			::	x(:)	x-values against which to differentiate
real(kind=dp),	intent(in)			::	y(:)	array of y-values, assuming $y(x)$ relation
real(kind=dp),	intent(out)			::	dy(size(y))	derivative of $y$ with respect to $x$, same size as input arrays
real(kind=dp),	intent(in),	optional		::	dxtol	optional tolerance for equally spaced arrays

Contents

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