Finite difference method

class tedeous.finite_diffs.First_order_scheme(term: list, nvars: int, axes_scheme_type: str)

Bases: object

Class for numerical scheme construction. Central o(h^2) difference scheme is used for ‘central’ points, forward (‘f’) and backward (‘b’) o(h) schemes are used for boundary points. ‘central’, and combination ‘f’,’b’ are corresponding to points_type. Args:

static finite_diff_shift(diff: list, axis: int, mode: str) → list

1st order points shift for the corresponding finite difference mode.

Parameters:

• diff – values of finite differences.

• axis – axis.

• mode – the finite difference mode (i.e., forward, backward, central).

Returns:

list with shifted points.

Return type:

diff_list

scheme_build() → list

Building first order (in terms of accuracy) finite-difference stencil.

Start from list of zeros where them numbers equal nvars. After that we move value in that axis which corresponding toterm. [0,0]->[[1,0],[-1,0]] it means that term was [0] (d/dx) and mode (scheme_type) is ‘central’.

Returns:

numerical scheme.

sign_order(h: float = 0.5) → list

Determines the sign of the derivative for the corresponding transformation from Finite_diffs.scheme_build().

From transformations above, we always start from +1 (1) Every +1 changes to ->[+1,-1] when order of differential rises [0,0] (+1) ->([1,0]-[-1,0]) ([+1,-1]) Every -1 changes to [-1,+1] [[1,0],[-1,0]] ([+1,-1])->[[1,1],[1,-1],[-1,1],[-1,-1]] ([+1,-1,-1,+1])

Parameters:

h – discretizing parameter in finite difference method (i.e., grid resolution for scheme).

Returns:

list, with signs for corresponding points.

class tedeous.finite_diffs.Second_order_scheme(term: list, nvars: int, axes_scheme_type: str)

Bases: object

Crank–Nicolson method. This realization only for boundary points.

scheme_build() → list

Scheme building for Crank-Nicolson variant, it’s identical to ‘scheme_build’ in first order method, but value is shifted by ‘second_order_shift’.

Returns:

numerical scheme list.

sign_order(h: float = 0.5) → list

Signs definition for second order schemes.

Parameters:

h – discretizing parameter in finite difference method (i.e., grid resolution for scheme).

Returns:

list, with signs for corresponding points.

class tedeous.finite_diffs.Finite_diffs(term: list, nvars: int, axes_scheme_type: str)

Bases: object

Class for numerical scheme choosing.

scheme_choose(scheme_label: str, h: float = 0.5)

Method for numerical scheme choosing via realized above.

Parameters:

• scheme_label – ‘2’- for second order scheme (only boundaries points), ‘1’ - for first order scheme.

• h – discretizing parameter in finite difference method (i.e., grid resolution for scheme).

Returns:

list where list[0] is numerical scheme and list[1] is signs.