brainpy.integrators.ode.DormandPrince

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

The Dormand–Prince method for ODEs.

The DOPRI method, is an explicit method for solving ordinary differential equations (Dormand & Prince 1980). The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error. For this reason, the Dormand–Prince method is more suitable when the higher-order solution is used to continue the integration, a practice known as local extrapolation (Shampine 1986; Hairer, Nørsett & Wanner 2008, pp. 178–179).

Its Butcher table is:

\[\begin{split}\begin{array}{l|llllll} 0 & \\ 1 / 5 & 1 / 5 & & & \\ 3 / 10 & 3 / 40 & 9 / 40 & & & \\ 4 / 5 & 44 / 45 & -56 / 15 & 32 / 9 & & \\ 8 / 9 & 19372 / 6561 & -25360 / 2187 & 64448 / 6561 & -212 / 729 & \\ 1 & 9017 / 3168 & -355 / 33 & 46732 / 5247 & 49 / 176 & -5103 / 18656 & \\ 1 & 35 / 384 & 0 & 500 / 1113 & 125 / 192 & -2187 / 6784 & 11 / 84 & \\ \hline & 35 / 384 & 0 & 500 / 1113 & 125 / 192 & -2187 / 6784 & 11 / 84 & 0 \\ & 5179 / 57600 & 0 & 7571 / 16695 & 393 / 640 & -92097 / 339200 & 187 / 2100 & 1 / 40 \end{array}\end{split}\]

References

[1]

https://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method

[2]

Dormand, J. R.; Prince, P. J. (1980), “A family of embedded Runge-Kutta formulae”, Journal of Computational and Applied Mathematics, 6 (1): 19–26, doi:10.1016/0771-050X(80)90013-3.

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

Methods

__init__(f[, var_type, dt, name, adaptive, ...])
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
B1
B2
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.