NSV, Euler Integration Confusion

Question

I'm trying to get my head around these integration methods and I'm thouroughly confused.

Here is the code:

public void update_euler(float timeDelta){
    vPos.y += vVelocity.y * timeDelta; 
    vVelocity.y += gravity.y * timeDelta;
}


public void update_nsv(float timeDelta){    
    vVelocity.y += gravity.y*timeDelta;
    vPos.y += vVelocity.y * timeDelta; 
}


public void onDrawFrame(GL10 gl) {      
    currentTime = System.currentTimeMillis();
    float timeDelta = currentTime - startTime;
    startTime = currentTime;
    timeDelta *= 1.0f/1000;;

//  update_RK4(timeDelta);
//  update_nsv(timeDelta);
//  update_euler(timeDelta);
//  update_velocity_verlet(timeDelta);
}

Firstly, I just want to make sure I've got these right.

I am simulating a perfectly elastic ball bouncing, so on the bounce I just reverse the velocity.

The Euler method, the ball bounces higher on each bounce. Is this due to an error in my code or is this due to the innacuracy of the method. I've read that with the Euler integration you lose energy over time. Well I'm gaining it and I don't know why.

The nsv method: I don't quite understand how this is different to the Eular method, but in any case the ball bounces lower on each bounce. It is losing energy which I've read isnt meant to happen with the nsv method. Why am I losing energy?

(The velocity verlet and RK4 methods are working as I'd expect them to).

I get the impression I'm lacking a fundamental bit of information on this subject, but I don't know what.

I do realise my timestep is lacking, and updating it to run the physics using a static timestep would stop me losing/gaining energy, but I am trying to understand what is going on.

Any help would be appreciated.

btilly btilly · Accepted Answer · 2011-06-23T00:35:13

To add another option to @Beta's answer, if you average the two methods, your error should disappear (except for issues around handling the actual bounce).

public void update_avg(float timeDelta){    
    vVelocity.y += gravity.y*timeDelta/2;
    vPos.y += vVelocity.y * timeDelta; 
    vVelocity.y += gravity.y*timeDelta/2;
}

What I'm doing here is updating the velocity to the average velocity over the interval, then updating the position based on that velocity, then updating the velocity to the velocity at the end of the interval.

If you have a more complicated scenario that you want to model, consider using the Runge-Kutta Method to solve differential equations of the form y' = f(x, y). (Note that here y can be a set of different variables. So in your case you'd have d(position, velocity)/dt = (velocity, -gravity). And the code I gave you works out to be the same as the second-order version of that method.

NSV, Euler Integration Confusion

3 Answers