cvode_solve

Arguments

f	a function, external, string or list, the right hand side of the differential equation.
y0	a double array: initial state of the ode.
options	a sequence of optional named arguments (see cvode options)
t	vector of time values used by the solver.
y	array of solution at time values in t
te,ye,ie	time of event, solution and index of event.
sol	MList of `_odeSolution` type

Description

cvode_solve computes the solution of real or complex ordinary different equations defined by . It is an interface to CVODE solver of Sundials library with default Adams method (BDF method can be chosen in options). The simplest call of cvode_solve is when tspan is a two component vector:

[t,y] = cvode_solve(f,[t0 tf],y0)

where y0 is the initial condition, t0, tf are the initial and final time, y is the array of solutions [y(t(1)),y(t(2)),...] at time values in t, where concatenation is done on next dimension if y0 is not a vector. The time values in t are those used by the solver to meet default relative and absolute estimated local error (which can be changed in options). In the simplest case the right hand side is computed by a Scilab function with two arguments, for example here f(t,y) = -y+t is coded as

function yprime = f(t,y)
yprime = t+y
end

See the section Functions for other possibilities. When t has more than two components:

[t,y] = cvode_solve(f,[t0 t1 ... tf],y0)

the solution is computed at prescribed points with the same precision as the two components syntax. However, the result may slightly differ (within chosen tolerance) since t1-t0 may give a different guess of the initial step used by the solver. Solver internal steps are the same and solution at user points is computed by continuous extension formulas.

When searching for (t,y) where some functions defined in options vanish (see the Events section below) the syntax

[t,y,te,ye,ie] = cvode_solve(f,tspan,y0,options)

allows to recover (t,y) values in te,ye and in ie the number(s) of the function.

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

sol = cvode_solve(f,[t0 tf],y0)

yields an MList of _odeSolution type, which can be used later as an interpolant, see the Solution output section.

Example

In the following example, we solve the Ordinary Differential Equation with the initial condition $y(0)=2,\,y$ and $\mu=1$ . Since the original equation is of second order we define the state of the equivalent first order equation as v=(y,y . The right hand side is computed by the function %cvode_vdp1 (defined in the Sundials module):

function vdot=%cvode_vdp1(t, v)
    mu = 1;
    vdot = [v(2); mu*(1-v(1)^2)*v(2)-v(1)]
end

[t,v] = cvode_solve(%cvode_vdp1, [0 10], [0; 2]);
plot(t,v)

Events

Agebraic equations to be solved simultaneously with the integration of the ODE can be specified by using the events option:

[t,y,te,ye,ie] = cvode_solve(f,tspan,y0,events = g)

where g in its simplest form is a Scilab function with prototype [eq,term,dir] = g(t,y), where eq(i)=0 when event i occurs, term is a boolean vector, with term(i)=%t value if integration has to be stopped when event i occurs. Vector dir allows to select event direction, with term(i) can take the values -1,1 if solution has to be decreasing or increasing or 0 if direction does not matter. If the corresponding behavior does not matter, dir or both term,dir outputs can be omitted in the function prototype. For example, in order to find all points in [0,10] where the solution of Van Der Pol equation veryfies y(t)=1 , we use the following code:

function eq=g(t, v)
    eq = v(1)-1;
end
[t,v,te,ve,ie] = cvode_solve(%cvode_vdp1, [0 10], [0; 2], events = g);
plot(te,ve(1,:),"or",t,v)

Solution output

The syntax sol = cvode_solve(f,[t0 tf],y0) yields an MList with fields sol.solver (the solver name, here "cvode"), sol.method ("adams" or "bdf"), sol.t (the solver time steps), sol.y (the solution at solver timesteps), sol.idata (a pointer to an internal CVODE object) and sol.stats, a structure hosting the solver statistics. When events have been considered additional fields sol.te, sol.ye, sol.ie are also present.

For an arbitrary time vector t, with values in the interval [t0,tf], y=sol(t) yields by costless interpolation the same approximation as if components of t where used in tspan. The MList sol can also be used to restart the solver and extend the solution for t>tf (see cvode_extend). 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_solve(%cvode_vdp1, [0 10], [0; 2]);
t = linspace(3,5,1000);
plot(t,sol(t))

Examples

// Complex ode

function out=crhs(t, y)
    out = 10*exp(2*%i*%pi*t)*y;
endfunction
[t,z] = cvode_solve(crhs,[0 5],1);
plot(t,real(z),t,imag(z))
legend "real(z)" "imag(z)"

Bibliography

A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers," ACM Transactions on Mathematical Software, 31(3), pp. 363-396, 2005. Also available as LLNL technical report UCRL-JP-200037.