Compact Finite Differences

class findiff.compact.CompactScheme(deriv, left: dict, right: list)

Represents compact finite difference as in Lele, J. Comp. Phys 103, 16-42 (1992)

A more appropriate name for the scheme would be “implicite finite differences”.

Normal, i.e. “explicit” finite differences express the n-th derivative as a linear combination neighboring function values:

\[f^{(n)}_i = \sum_k c_k f_{i+k}\]

The compact/implicit finite difference scheme, however uses:

sum_k alpha_k f^{(n)}_{i+k} = sum_k c_k f_{i+k}

So, in order to apply implicit finite differences, one has to solve a (sparse) linear equation system.

The big advantage of the implicit scheme is that for the same accuracy order, one needs fewer neighboring points than in the explicit schemes.

Initializes a CompactScheme instance.

The compact/implicit finite difference scheme is defined by:

\[\sum_k \alpha_k f^{(n)}_{i+k} = \sum_k c_k f_{i+k}\]

Args:

deriv: int

The order of the derivative

left: dict

Defines the alphas in the formula above. Keys: k, Values: alpha_k

right: list|tuple

Which neighboring points to use (the k’s in the formula above)