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Solution output (ODE and DAE solvers)

Specialized output of solvers

Syntax

sol = solver(..., options)
solext = solver(sol, tfinal, options)
y = sol(t)
[y,yd] = sol(t)
... = sol(t,i)

Arguments

sol, solext

MList of _odeSolution type

tfinal

final time of extended solution

options

a sequence of optional named arguments (see sundials options)

t

a vector of time values

y,yd

The matrices of interpolated solution and its derivative at times values given in t

i

the component of the solution to interpolate. If the state of the equation is not a vector then i indexes the state as a flatened vector.

Description

When a solution has to be further evaluated at arbitrary points which are not known in advance, then the syntax

sol = solver(..., options)

yields a MList with fields:

solver

the solver name, "arkode", "cvode" or "ida"

method

the name of the method used by the solver. For example cvode can use "ADAMS" or "BDF", ida uses only "BDF" and the list of arkode methods is given in its dedicated help page.

interpolation

a string giving the interpolation method used by the solver to compute the solution between solver steps. The solvers arkode and cvode interpolation is based on the basis vectors used by the solver and thus the interpolation method is denoted "native". arkode can use Hermite or Lagrange interpolation and the polynomial degree can be chosen when calling the solver, hence the string is "Hermite(p)" or "Lagrange(p)" where p is the actual degree.

linearSolver

the linear solver used by the method, if applicable. Can be "none" when using the Adams method with cvode or an explicit method with arkode. When using an implicit method relying on Newton method for the iterations, and when the linear solver is matrix based, i.e. implements a direct method, this string can be "dense", "band" or "sparse" depending on the format of the Jacobian provided by the user.

nonLinearSolver

the nonlinear solver used by the method, which can be "none" for explicit methods, "fixedPoint" when functional iterations are used or "Newton", when the Newton method is used, both when using an implicit method (e.g. BDF or any DIRK method).

rtol,atol

the relative and absolute tolerance used by the solver to achieve adaptive time stepping.

t,y,yp

vector of time points used by the solver and solution at time points (yp is defined when using ida)

te,ye,ype,ie

time of event, solution and index of event

manager

pointer to the internal OdeManager object

stats

a structure gathering the solver statistics:

nStepsthe number of internal solver steps
nRhsEvalsthe number of right hand side evaluations
nRhsEvalsFDthe number of right hand side evaluations that have been used to approximate the Jacobian
nJacEvalsthe number of Jacobian evaluations, when this one has been given by the user
nEventEvalsthe number event function evaluations
nLinSolvethe number of linear systems that have been solved
nRejStepsthe number of solver steps that have been rejected
nNonLinitersthe number of iterations of the nonlinear method used by the solver
nNonLinCVFailsthe number of times that the nonlinear method failed to converge
orderthe order of the method for arkode or the vector of successive orders for cvode and ida
hInithe initial step used by the solver. Can match the initial step suggested by the "h0" option
hLastthe step used at the previous solver time point
hCurthe step that will be tried for the next solver iteration
etimeelapsed time by the solver

Solution interpolation and extension

For an arbitrary time vector t, with values in the interval [t0,tf] the following syntax

y = sol(t)
[y,yd] = sol(t)
... = sol(t,i)

yields by costless interpolation in y and yd the same approximation as if components of t where used in the tspan argument in the solver call. The derivative of solution can also be obtained with [y,yd]=sol(t) and a particular component i of the solution is obtained with y=sol(t,i). For example, the solution of the Van Der Pol equation can be refined in [3,5] with the following code:

sol = cvode(%SUN_vdp1, [0 10], [0; 2]);
t = linspace(3,5,1000);
plot(t,sol(t))

The syntax

solext = solver(sol, tfinal, options)

extends the solution by restarting the solver from initial time sol.t($) and initial condition sol.y(:,$) (you cannot use a different solver or the same solver with a different method). This extension can be typically used when you know in advance that the solution is not differentiable at some time point. By stopping then restarting the solver at time of discontinuity you optimize solver effort by avoiding very small time steps.

In the options sequence you can change almost all (but not the solver method) previously used options used in the call of the solver which yielded sol. The right hand side or the residual function and the initial conditions can be overriden by using the specific rhs, res, y0 or yp0 options, respectively. Here is an example where we extend the solution of the Van Der Pol equation:

sol = arkode(%SUN_vdp1, [0 5], [0; 2]);
solext = arkode(sol, 10);
t = linspace(0,5,500);
plot(t,sol(t,1))
t = linspace(5,10,500);
plot(t,solext(t,1),'r')

See also

  • arkode — SUNDIALS ordinary differential equation additive Runge-Kutta solver
  • cvode — SUNDIALS ordinary differential equation solver
  • ida — SUNDIALS differential-algebraic equation solver
  • User functions — Coding user functions used by SUNDIALS solvers
  • Options (ODE and DAE solvers) — Changing the default behavior of solver
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Last updated:
Mon Jun 17 17:53:22 CEST 2024