confusing ode suits for solving discontinuous odes

Hi all! when I use ode suits with event option to deal with the problem about discontinuous odes, here is a ball bouncing on the ground, why are the points of the solution not evenly distributed.

Below is my code.

clc
clear
tic
 %options=odeset('Events',@Events,'AbsTol',1e-8,'RelTol',1e-8);
 options=odeset('Events',@Events);
 y0 = [1;0];
 [tout,yout]=ode45(@Tq,[0,7], y0,options);
subplot(1,2,1)
plot(tout,yout(:,1),'k')
subplot(1,2,2)
plot(tout,yout(:,1),'k.')
toc

function f=Tq(~,y)
      if y(1)>0
          u=0;
      else
          u=30*y(2)+1e5*y(1);
      end
      f=[y(2); -9.81-u];
end
 
 function [g,isterminal,direction]=Events(t,y)
    g=y(1);
    isterminal=0;
    direction=0;
 end

[Bobby Fischer]

It was nice to see the 'Events' part which I didn't know about.

I could be wrong, but it seems to me one has two options: either you let ode45 see the 'events' part and then don't get equally spaced points, or you tell ode45 the specific times at which you want the calculations to be made, but then don't get to be more acute at the points which would need it the most. In the second case you can always give a lot of points and that would get things done, but that would be less elegant.

I don't know what's your opinion about this.

Answer 1

[Mischa Kim]

Hi, this is because the integrator, ode45, adjusts the integration step size. Broadly speaking where the dynamics is more complex the integrator typically decreases the step size (e.g. when the ball hits the ground), when the dynamics is less complex the integrator tries to increase step size and thereby reduce compute time. See the documentation here for more information. If equally spaced data points are important to you, you can adjust the tspan vector accordingly.

Answer 2

[汉武 沈]

Okay, grateful for your help!