brainpy.integrators.ode.Heun2

class brainpy.integrators.ode.Heun2(f, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)

Heun’s method for ODEs.

This method is named after Karl Heun 1. It is also known as the explicit trapezoid rule, improved Euler’s method, or modified Euler’s method.

Given ODEs with a given initial value,

\[y'(t) = f(t,y(t)), \qquad y(t_0)=y_0,\]

the two-stage Heun’s method is formulated as:

\[\tilde{y}_{n+1} = y_n + h f(t_n,y_n)\]

\[y_{n+1} = y_n + \frac{h}{2}[f(t_n, y_n) + f(t_{n+1},\tilde{y}_{n+1})],\]

where \(h\) is the step size and \(t_{n+1}=t_n+h\).

Therefore, the Butcher tableau of the two-stage Heun’s method is:

\[\begin{split}\begin{array}{c|cc} 0.0 & 0.0 & 0.0 \\ 1.0 & 1.0 & 0.0 \\ \hline & 0.5 & 0.5 \end{array}\end{split}\]

Geometric interpretation

In the brainpy.integrators.ode.midpoint(), we have already known Euler method has big estimation error because it uses the line tangent to the function at the beginning of the interval \(t_n\) as an estimate of the slope of the function over the interval \((t_n, t_{n+1})\).

In order to address this problem, Heun’s Method considers the tangent lines to the solution curve at both ends of the interval (\(t_n\) and \(t_{n+1}\)), one (\(f(t_n, y_n)\)) which underestimates, and one (\(f(t_{n+1},\tilde{y}_{n+1})\), approximated using Euler’s Method) which overestimates the ideal vertical coordinates. The ideal point lies approximately halfway between the erroneous overestimation and underestimation, the average of the two slopes.

\[\begin{split}\begin{aligned} {\text{Slope}}_{\text{left}}=&f(t_{n},y_{n}) \\ {\text{Slope}}_{\text{right}}=&f(t_{n}+h,y_{n}+hf(t_{n},y_{n})) \\ {\text{Slope}}_{\text{ideal}}=&{\frac {1}{2}}({\text{Slope}}_{\text{left}}+{\text{Slope}}_{\text{right}}) \end{aligned}\end{split}\]

References

1

Süli, Endre, and David F. Mayers. An Introduction to Numerical Analysis. no. 1, 2003.

__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])	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, ...)
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)	New in version 2.3.1.
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.