An implementation of solve_ivp (ODE solver)

Question

This is my first post on stack overflow. So I came across this example of solve_ivp solver in SciPy's documentation. The problem being solved is the following:

Cannon fired upward with terminal event upon impact. The terminal and direction fields of an event are applied by monkey patching a function. Here y[0] is position and y[1] is velocity. The projectile starts at position 0 with velocity +10. Note that the integration never reaches t=100 because the event is terminal.

The code in the documentation is the following:

>>> def upward_cannon(t, y): return [y[1], -0.5]
>>> def hit_ground(t, y): return y[0]
>>> hit_ground.terminal = True
>>> hit_ground.direction = -1
>>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)
>>> print(sol.t_events)
[array([40.])]
>>> print(sol.t)
[0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
 1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01]

I have used this solver for solving other differential equations. Further, I understand the usage terminal and direction fields. But for this example only, I am not able to understand how the function upward_cannon() is working by using return [y[1],-0.5] .The output corresponding to print(sol.y) is the following:

[[ 0.00000000e+00  9.99897510e-04  1.09985977e-02  1.10958105e-01
   1.10780373e+00  1.08013149e+01  8.02419261e+01 -1.42108547e-14]
 [ 1.00000000e+01  9.99995000e+00  9.99945005e+00  9.99445055e+00
   9.94445555e+00  9.44450555e+00  4.44500550e+00 -1.00000000e+01]]

Since there is no underlying differential equation mentioned in the code, how is the solver producing the above values for y ? I know return [y[1],-0.5] is doing something but I am not able to explain what it is doing.

Answer 1

You wrote

Since there is no underlying differential equation mentioned in the code...

It is not explicitly mentioned, but in fact, there is a differential equation. Let h(t) be the height of the cannonball. The differential equation is

h''(t) = -0.5

That is, the acceleration is constant and in the downward direction. This is Newton's second law , with the assumption of constant gravitational force. The example assumes that the ratio of the gravitational constant and the mass is 0.5.

The initial conditions are

h(0) = 0,  h'(0) = 10

The equation h''(t) = -0.5 is a (trivial!) second order differential equation. We could solve it with plain old definite integration, but for the sake of the example, we use an ODE solver. To use solve_ivp (or any of the other ODE solvers in SciPy), we must convert the second order DE to a system of first order equations. Let y0(t) = h(t) and y1(t) = h'(t). Then

y0'(t) = h'(t) = y1(t)
y1'(t) = h''(t) = -0.5

That is the system that is implemented in

def upward_cannon(t, y):
    return [y[1], -0.5]