brainpy.integrators.ode.RK3

class brainpy.integrators.ode.RK3(f, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)

Classical third-order Runge-Kutta method for ODEs.

For the given initial value problem \(y'(x) = f(t,y);\, y(t_0) = y_0\), the third order Runge-Kutta method is given by:

\[y_{n+1} = y_n + 1/6 ( k_1 + 4 k_2 + k_3),\]

where

\[\begin{split}k_1 = h f(t_n, y_n), \\ k_2 = h f(t_n + h / 2, y_n + k_1 / 2), \\ k_3 = h f(t_n + h, y_n - k_1 + 2 k_2 ),\end{split}\]

where \(t_n = t_0 + n h.\)

Error term \(O(h^4)\), correct up to the third order term in Taylor series expansion.

The Taylor series expansion is \(y(t+h)=y(t)+\frac{k}{6}+\frac{2 k_{2}}{3}+\frac{k_{3}}{6}+O\left(h^{4}\right)\).

The corresponding Butcher tableau is:

\[\begin{split}\begin{array}{c|ccc} 0 & 0 & 0 & 0 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 1 & -1 & 2 & 0 \\ \hline & 1 / 6 & 2 / 3 & 1 / 6 \end{array}\end{split}\]

__init__(f, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)

Methods

__init__(f[, var_type, dt, name, show_code, ...])
build()
cpu()	Move all variable into the CPU device.
cuda()	Move all variables into the GPU device.
load_state_dict(state_dict[, warn, compatible])	Copy parameters and buffers from state_dict into this module and its descendants.
load_states(filename[, verbose])	Load the model states.
nodes([method, level, include_self])	Collect all children nodes.
register_implicit_nodes(*nodes[, node_cls])
register_implicit_vars(*variables[, var_cls])
save_states(filename[, variables])	Save the model states.
set_integral(f)	Set the integral function.
state_dict()	Returns a dictionary containing a whole state of the module.
to(device)	Moves all variables into the given device.
tpu()	Move all variables into the TPU device.
train_vars([method, level, include_self])	The shortcut for retrieving all trainable variables.
tree_flatten()	Flattens the object as a PyTree.
tree_unflatten(aux, dynamic_values)	Unflatten the data to construct an object of this class.
unique_name([name, type_])	Get the unique name for this object.
vars([method, level, include_self, ...])	Collect all variables in this node and the children nodes.

Attributes

A
B
C
arguments	All arguments when calling the numer integrator of the differential equation.
dt	The numerical integration precision.
integral	The integral function.
name	Name of the model.
neutral_delays	neutral delays.
parameters	The parameters defined in the differential equation.
state_delays	State delays.
variables	The variables defined in the differential equation.