Dopri5

Explicit Runge-Kutta method of order (4)5 with step-size control and continuous output. Based on the method by Dormand and Prince.

Based on the FORTRAN code DOPRI5 by E.Hairer and G.Wanner, which can be found here: http://www.unige.ch/~hairer/software.html

Details about the implementation (FORTRAN) can be found in the book,:

Solving Ordinary Differential Equations I,
Nonstiff Problems

Authors: E. Hairer, S. P. Norsett and G. Wanner
Springer-Verlag, ISBN: 3-540-56670-8

Support

• State events (root funtions) : True
• Step events (completed step) : True
• Time events : True

Usage

Import the solver together with the correct problem:

from assimulo.solvers import Dopri5
from assimulo.problem import Explicit_Problem

Define the problem, such as:

def rhs(t, y): #Note that y are a 1-D numpy array.
    yd = -1.0
    return N.array([yd]) #Note that the return must be numpy array, NOT a scalar.

y0 = [1.0]
t0 = 1.0

Create a problem instance:

mod = Explicit_Problem(rhs, y0, t0)

Note

For complex problems, it is recommended to check the available examples and the documentation in the problem class, Explicit_Problem. It is also recommended to define your problem as a subclass of Explicit_Problem.

Warning

When subclassing from a problem class, the function for calculating the right-hand-side (for ODEs) must be named rhs and in the case with a residual function (for DAEs) it must be named res.

Create a solver instance:

sim = Dopri5(mod)

Modify (optionally) the solver parameters.

Parameters:

• atol Defines the absolute tolerance(s) that is to be used by the solver.
• backward Specifies if the simulation is done in reverse time.
• beta Option for stabilized step-size control.
• clock_step Specifies if the elapsed time of an integrator step should be timed or not.
• display_progress This option actives output during the integration in terms of that the current integration is periodically printed to the stdout.
• fac1 Parameters for step-size selection.
• fac2 Parameters for step-size selection.
• inith This determines the initial step-size to be used in the integration.
• maxh Defines the maximal step-size that is to be used by the solver.
• maxsteps The maximum number of steps allowed to be taken to reach the final time.
• num_threads This options specifies the number of threads to be used for those solvers that supports it.
• report_continuously This options specifies if the solver should report the solution continuously after steps.
• rtol Defines the relative tolerance that is to be used by the solver.
• safe The safety factor in the step-size prediction.
• store_event_points This options specifies if the solver should save additional points at the events, \(t_e^-, t_e^+\).
• time_limit This option can be used to limit the time of an integration.
• verbosity This determines the level of the output.

Methods:

• Dopri5.interpolate

Simulate the problem:

t, y = Dopri5.simulate(tfinal, ncp, ncp_list)

Information:

• Dopri5.get_options() Returns the current solver options.
• Dopri5.get_supports() Returns the functionality which the solver supports.
• Dopri5.get_statistics() Returns the run-time statistics (if any).
• Dopri5.get_event_data() Returns the event information (if any).
• Dopri5.print_event_data() Prints the event information (if any).
• Dopri5.print_statistics() Prints the run-time statistics for the problem.