araujo88/finite_difference

Computes finite difference matrices for the first and second derivative up to sixth order, including compact schemes

PythonGPL-3.0

finite_difference

Computes finite difference matrices for the first and second derivative up to sixth order, including compact finite-difference schemes such as the fourth-order Padé scheme and sixth-order Lele scheme (S. K. Lele. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103(1):16–42, November 1992).

Methods available

2nd order
4th order
6th order
Padé (4th order compact scheme)
Lele (6th order compact scheme)

Examples

Example 1 - derivatives_example.py

This example is to demonstrate the perfomance of the different methods available. The function used in this example is given by:

$f(x) = \e^{x^2}$

which has the following derivatives:

$f'(x) = 2 x \e^{x^2}$

${f''(x) = 2 \e^{x^2} (1 + 2x^2)}$

Relative error (in log scale) obtained for the first derivative:

Relative error (in log scale) obtained for the second derivative:

Example 2 - 2D_first_derivatives_example.py

The function shown in the example files is given by:

$f(x,y) = \sin x \ \sin y$

The partial derivatives are given by:

$\frac{\partial f}{\partial x} = \cos x \ \sin y$

$\frac{\partial f}{\partial y} = \sin x \ \cos y$

The images below show the absolute error in log scale for both the partial derivatives in x and y direction.

Second order

Fourth order

Sixth order

Padé fourth order

Lele sixth order