rk4 module

Define Runge-Kutta integraiton function.

rk4(u, du, x, dx)

Compute variation of u between x and x+dx.

Use 4-th order Runge-Kutta method.

Note

This is a slightly modified version of the RK. The k_i are proportional to du instead of du/dx.

Parameters:

• u (ndarray[tuple[Any, ...], dtype[double]]) – Vector to integrate.

• du (Callable[[float, ndarray[tuple[Any, ...], dtype[double]]], ndarray[tuple[Any, ...], dtype[double]]]) – Gives the variation of u components with x.

• x (float) – Where u is known.

• dx (float) – Integration step.

Return type:

ndarray[tuple[Any, ...], dtype[double]]

Returns:

Variation of u between x and x+dx.

rk4_2d(u, v, delta, x, dx)

Compute variation of u and v between x and x+dx.

Use 4-th order Runge-Kutta method. u and v are scalar. This version is rk4() but with two dimensions only.

Note

This is a slightly modified version of the RK. The k_i are proportional to delta_u instead of du_dz.

Parameters:

• u (float) – First variable to be integrated.

• v (float) – Second variable to be integrated.

• delta (Callable[[float, float, float], tuple[float, float]]) – Takes in x, u, v and return variation of u and v.

• x (float) – Where u and v are known.

• dx (float) – Integration step.

Return type:

tuple[float, float]

Returns:

Variation of u and v between x and x+dx.