Solving

After creating your Temporal Variables, you can simulate your system using:

vip.solve(t_end)

This will integrate the system from time t = 0 to t = t_end.

Choosing Time Points

The vip.solve() function supports different ways to define the time points of the simulation:

• Fixed time step (default):
  Specify the time_step argument to produce outputs at regular intervals.
• Automatic time points:
  Leave time_step=None to output only the time points that the solver chooses adaptively.
• Custom time points:
  Provide a list of time values using the t_eval argument.

If your system includes crossing detections, you can disable automatic insertion of crossing time points by setting:

include_crossing_times=False

Choosing a Solving Method

You can choose among several integration methods provided by SciPy by setting the method argument:

Method	When to Use	Notes
"RK45" (default)	General-purpose for smooth, non-stiff problems	Fast, accurate, and a good default choice.
"RK23"	Same as RK45, but with lower-order accuracy	Takes more steps, useful if you need dense sampling.
"DOP853"	For very smooth problems requiring high precision	Higher-order method. Slower but more accurate for some problems.
"Radau"	For stiff problems	Implicit solver, stable for stiff systems but slower.
"BDF"	Also for stiff systems, especially long simulations	Multi-step implicit solver. Efficient for large or slowly evolving systems.
"LSODA"	Unsure whether the system is stiff or not?	Automatically switches between stiff and non-stiff solvers. Great for mixed or unknown cases.

👉 See SciPy’s solve_ivp() documentation for technical details on each method.

Controlling Precision

Solvers approximate the true solution. You can control the precision using the following parameters:

• rtol: Relative tolerance (default: 1e-3)
• atol: Absolute tolerance (default: 1e-6)

The solver ensures the local error stays below:

atol + rtol * abs(y)

You can also set the max_step argument to limit how far the solver can go in one integration step, which can improve precision.

note

Difference between max_step and time_step:

• max_step controls the maximum internal step size used by the solver.
• time_step defines how often the solution is output. It does not modify how the integration method works. However, events are evaluated at each time step, so a smaller time step may detect higher frequency events.

Other Options

• plot=False:
  Disables the automatic plotting of variables marked with .to_plot().

• verbose=True:
  Prints useful information about the solver, performance, and triggered events.