Numerical integration, Runge-Kutta, RK4 in game design

Question

Nearly every game tends to use some of a game loop. Gafferongames has a great article on how to make a well designed game loop: http://gafferongames.com/game-physics/fix-your-timestep/

In his code, he uses integrate( state, t, deltaTime );, where I believe state contains position, velocity, and acceleration of the object. He uses RK4 to integrate it from t to t+deltaTime.

My question is, why use a numerical integration technique like RK4, when you can use kinematics equations (here) to find the exact value?

These equations work when acceleration is constant. It seems rare that you would have a changing acceleration within a timestep. It seems like RK4 is a lower performance, lower accuracy, more complex solution.

Edit: I think you could add a "jerk" value to objects and still find exact expressions for acceleration, velocity, and displacement, if you really wanted to.

Edit 2: Well, I did not read his "Integration Basics" article too carefully. I think he's modelling a damper and spring, which do cause non-constant acceleration within a timestep.

Answer 1

As soon as you add things that many game designers want, like (velocity dependent) drag, position dependent forces, etc. the equations are no longer solvable exactly.

So, if you're happy to limit your forces to those the kinematic equation can handle, then go with it. If you want something flexible, then numerical integration is the only way to go.

Note: If you treat the forces as constant over a time interval when they are not really constant - then you are actually using a form of numerical integration. And it is an inaccurate form of integration too. So why not use a tried and proven numerical method instead? RK4 is one of many such methods.

Answer 2

Approximating acceleration (derivatives, really) as constant within a time step is how numerical integration methods work. When the derivatives are not constant, you need to consider what sort of error you introduce by treating them as constant.

Imagine breaking a time range T up into N equal steps of width h=T/N. Now integrate the dynamical equations stepwise. With RK4, the local error per-step is O(h^5) giving a global error of O(h^4).

Using the kinematical equations as you propose, we can assess the error by considering the Taylor expansion of the position, keeping terms to second order. The position will have error of O(h^3) introduced at each step, corresponding to where you truncate the expansion. This gives local error O(h^3) and global error O(h^2).

Based on the asymptotic error, the error from RK4 goes to zero much more rapidly than does the kinematical equations. It's more accurate. RK4 obtains a very nice accuracy obtained for the number of function evaluations that need to be done.