DAE Solvers

class scikits_odes.dae.dae(integrator_name, eqsres, **options)

A generic interface class to differential algebraic equations.

Define equation res = G(t,y,y’) which can eg be G = f(y,t) - A y’ when solving A y’ = f(y,t), and where (optional) jac is the jacobian matrix of the nonlinear system see fortran source code), so d res/dy + scaling * d res/dy’ or d res/dy depending on the backend.

Parameters:

• integrator_name ('ida', 'ddaspk' or 'lsodi') – The integrator solver to use.

• eqsres (residual function) –

  Residual of the DAE. The signature of this function depends on the solver used, see the solver documentation for details. Generally however, you can assume the following signature to work:

  eqsres(x, y, yprime, return_residual)

  with x : independent variable, eg the time, float y : array of n unknowns in x yprime : dy/dx array of n unknowns in x, dimension = dim(y) return_residual: array that must be updated with the value of the residuals, so G(t,y,y’). The dimension is equal to dim(y) return value: integer, 0 for success. It is not guaranteed that a solver takes this status into account

  Some solvers will allow userdata to be passed to eqsres, or optional formats that are more performant.

• options (mapping) – Additional options for initialization, solver dependent See set_options method of the integrator_name you selected for details.

See also

odeint

an ODE integrator with a simpler interface based on lsoda from ODEPACK

ode

class around vode ODE integrator

Notes

Possible future solvers

ddaskr: Not included, starting hints: http://osdir.com/ml/python.f2py.user/2005-07/msg00014.html Modified Extended Backward Differentiation Formulae (MEBDF): Not included. Fortran codes: http://www.ma.ic.ac.uk/~jcash/IVP_software/readme.html

Examples

DAE arise in many applications of dynamical systems, as well as in discritisations of PDE (eg moving mesh combined with method of lines). As an easy example, consider the simple oscillator, which we write as G(y,y’,t) = 0 instead of the normal ode, and solve as a DAE.

>>> from __future__ import print_function
>>> from numpy import cos, sin, sqrt
>>> k = 4.0
>>> m = 1.0
>>> initx = [1, 0.1]
>>> initxp = [initx[1], -k/m*initx[0]]
>>> def reseqn(t, x, xdot, result):
    ... # we create residual equations for the problem
    ... result[0] = m*xdot[1] + k*x[0]
    ... result[1] = xdot[0] - x[1]
>>> from scikits_odes import dae
>>> solver = dae('ida', reseqn)
>>> result = solver.solve([0., 1., 2.], initx, initxp)

init_step(t0, y0, yp0, y_ic0_retn=None, yp_ic0_retn=None)

Initializes the solver and allocates memory. It is not needed to call this method if solve is used to compute the solution. In the case step is used, init_step must be called first.

Parameters:

• t0 (number) – initial time

• y0 (list/array) – initial condition for y

• yp0 (list/array) – initial condition for yp

• y_ic0 (numpy array) – (optional) returns the calculated consistent initial condition for y

• yp_ic0 (numpy array) – (optional) returns the calculated consistent initial condition for y derivated.

Returns:

• old_api is False (namedtuple) – namedtuple with the following attributes

  Field	Meaning
  flag	An integer flag (StatusEnumXXX)
  values	Named tuple with entries t and y and ydot. y will correspond to y_retn value and ydot to yp_retn!
  errors	Named tuple with entries t and y and ydot
  roots	Named tuple with entries t and y and ydot
  tstop	Named tuple with entries t and y and ydot
  message	String with message in case of an error

• old_api is True (tuple) – tuple with the following elements in order

  Field	Meaning
  flag	status of the computation (successful or error occurred)
  t_out	time, where the solver stopped (when no error occurred, t_out == t)

set_options(**options)

Set specific options for the solver. See the solver documentation for details.

Calling set_options a second time, normally resets the solver.

solve(tspan, y0, yp0)

Runs the solver.

Parameters:

• tspan (list/array) – A list of times at which the computed value will be returned. Must contain the start time as first entry.

• y0 (list/array) – list array of initial values

• yp0 (list/array) – list array of initial values of derivatives

Returns:

• old_api is False (namedtuple) – namedtuple with the following attributes

  Field	Meaning
  flag	An integer flag (StatusEnumXXX)
  values	Named tuple with entries array t and array y and array ydot. y will correspond to y_retn value and ydot to yp_retn!
  errors	Named tuple with entries t and y and ydot of error
  roots	Named tuple with entries array t and array y and array ydot
  tstop	Named tuple with entries array t and array y and array ydot
  message	String with message in case of an error

• old_api is True (tuple) – tuple with the following elements in order

  Field	Meaning
  flag	indicating return status of the solver
  t	numpy array of times at which the computations were successful
  y	numpy array of values corresponding to times t (values of y[i, :] ~ t[i])
  yp	numpy array of derivatives corresponding to times t (values of yp[i, :] ~ t[i])
  t_err	float or None - if recoverable error occurred (for example reached maximum number of allowed iterations), this is the time at which it happened
  y_err	numpy array of values corresponding to time t_err
  yp_err	numpy array of derivatives corresponding to time t_err

step(t, y_retn=None, yp_retn=None)

Method for calling successive next step of the IDA solver to allow more precise control over the IDA solver. The ‘init_step’ method has to be called before the ‘step’ method.

A step is done towards time t, and output at t returned. This time can be higher or lower than the previous time. If option ‘one_step_compute’==True, and the solver supports it, only one internal solver step is done in the direction of t starting at the current step.

If old_api=True, the old behavior is used: if t>0.0 then integration is performed until this time and results at this time are returned in y_retn; else if if t<0.0 only one internal step is performed towards time abs(t) and results after this one time step are returned.

Parameters:

• t (number)

• y_retn (numpy array (ndim = 1) or None.) – (Needs to be preallocated) If not None, will be filled with y at time t. If None y_retn is not used.

• yp_retn (numpy array (ndim = 1) or None.) – (Needs to be preallocated) If not None, will be filled with derivatives of y at time t. If None yp_retn is not used.

Returns:

• old_api is False (namedtuple) – namedtuple with the following attributes

  Field	Meaning
  flag	An integer flag (StatusEnumXXX)
  values	Named tuple with entries t and y and ydot. y will correspond to y_retn value and ydot to yp_retn!
  errors	Named tuple with entries t and y and ydot
  roots	Named tuple with entries t and y and ydot
  tstop	Named tuple with entries t and y and ydot
  message	String with message in case of an error

• old_api is True (tuple) – tuple with the following elements in order

  Field	Meaning
  flag	status of the computation (successful or error occurred)
  t_out	time, where the solver stopped (when no error occurred, t_out == t)