brainpy.integrators.ode.RK4#
- class brainpy.integrators.ode.RK4(f, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)[source]#
Classical fourth-order Runge-Kutta method for ODEs.
For the given initial value problem of
\[{\frac {dy}{dt}}=f(t,y),\quad y(t_{0})=y_{0}.\]The fourth-order RK method is formulated as:
\[\begin{split}\begin{aligned} y_{n+1}&=y_{n}+{\frac {1}{6}}h\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right),\\ t_{n+1}&=t_{n}+h\\ \end{aligned}\end{split}\]for \(n = 0, 1, 2, 3, \cdot\), using
\[\begin{split}\begin{aligned} k_{1}&=\ f(t_{n},y_{n}),\\ k_{2}&=\ f\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{1}}{2}}\right),\\ k_{3}&=\ f\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{2}}{2}}\right),\\ k_{4}&=\ f\left(t_{n}+h,y_{n}+hk_{3}\right). \end{aligned}\end{split}\]Here \(y_{n+1}\) is the RK4 approximation of \(y(t_{n+1})\), and the next value (\(y_{n+1}\)) is determined by the present value (\(y_{n}\)) plus the weighted average of four increments, where each increment is the product of the size of the interval, \(h\), and an estimated slope specified by function \(f\) on the right-hand side of the differential equation.
\(k_{1}\) is the slope at the beginning of the interval, using \(y\) (Euler’s method);
\(k_{2}\) is the slope at the midpoint of the interval, using \(y\) and \(k_{1}\);
\(k_{3}\) is again the slope at the midpoint, but now using \(y\) and \(k_{2}\);
\(k_{4}\) is the slope at the end of the interval, using \(y\) and \(k_{3}\).
The RK4 method is a fourth-order method, meaning that the local truncation error is on the order of (\(O(h^{5}\)), while the total accumulated error is on the order of (\(O(h^{4}\)).
The corresponding Butcher tableau is:
\[\begin{split}\begin{array}{c|cccc} 0 & 0 & 0 & 0 & 0 \\ 1 / 2 & 1 / 2 & 0 & 0 & 0 \\ 1 / 2 & 0 & 1 / 2 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ \hline & 1 / 6 & 1 / 3 & 1 / 3 & 1 / 6 \end{array}\end{split}\]References
- __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.