1. Integration Techniques
Marq Singer

2. Essential Math for Games
Integrators
Solve “initial value problem” for ODEs
Used Euler’s method in previous talk
But not the only way to do it
Are other, more stable ways
Essential Math for Games

3. Essential Math for Games
The Problem
Physical simulation with force dependant on position or velocity
Start at x0, v0
Only know:
Essential Math for Games

4. Essential Math for Games
The Solution
Do an iterative solution
Start at some initial value
Ideally follow a step-by-step (or stepwise) approximation of the function
Essential Math for Games

5. Euler’s Method (review)
Idea: we have the slope (x or v)
Follow slope to find next values of x or v
Start with x0, v0, time step h
Essential Math for Games

6. Essential Math for Games
Euler's Method
Step across vector field of functions
Not exact, but close x x0 x1 x2 t
Essential Math for Games

7. Euler’s Method (cont’d)
Has problems
Expects the slope at the current point is a good estimate of the slope on the interval
Approximation can drift off the actual function – adds energy to system!
Gets worse the farther we get from known initial value
Especially bad when time step gets larger
Essential Math for Games

8. Euler’s Method (cont’d)
Example of drift x x0 x2 t x1
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9. Essential Math for Games
Stiffness
Running into classic problem of stiff equations
Have terms with rapidly decaying values
Larger decay = stiffer equation = need smaller h
Often seen in equations with stiff springs (hence the name)
Essential Math for Games

10. Essential Math for Games
Midpoint Method
Take two approximations
Approximate at half the time step
Use slope there for final approximation x x0 h x1
h/2
x0.5 t
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11. Essential Math for Games
Midpoint Method
Writing it out:
Can still oscillate if h is too large
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12. Essential Math for Games
Runga-Kutta
Use weighted average of slopes across interval
How error-resistant indicates order
Midpoint method is order two
Usually use Runga-Kutta Order Four, or RK4
Essential Math for Games

13. Essential Math for Games
Runga-Kutta (cont’d)
Better fit, good for larger time steps
Expensive -- requires many evaluations
If function is known and fixed (like in physical simulation) can reduce it to one big formula
But for large timesteps, still have trouble with stiff equations
Essential Math for Games

14. Essential Math for Games
Implicit Methods
Explicit Euler methods add energy
Implicit Euler removes it
Use new velocity, not current
E.g. Backwards Euler:
Better for stiff equations
Essential Math for Games

15. Essential Math for Games
Implicit Methods
Result of backwards Euler
Solution converges more slowly
But it converges! x x0 x1 x2 t
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16. Essential Math for Games
Implicit Methods
How to compute x'i+1 or v'i+1?
Derive from formula (most accurate)
Compute using explicit method and plug in value (predictor-corrector)
Solve using linear system (slowest, most general)
Essential Math for Games

17. Essential Math for Games
Implicit Methods
Example of predictor-corrector:
Essential Math for Games

18. Essential Math for Games
Implicit Methods
Solving using linear system:
Resulting matrix is sparse, easy to invert
Essential Math for Games

19. Essential Math for Games
Verlet Integration
Velocity-less scheme
From molecular dynamics
Uses position from previous time step
Stable, but not as accurate
Good for particle systems, not rigid body
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20. Essential Math for Games
Verlet Integration
Others:
Leapfrog Verlet
Velocity Verlet
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21. Essential Math for Games
Multistep Methods
Previous methods used only values from the current time step
Idea: approximation drifts more the further we get from initial value
Use values from previous time steps to calculate next one
Anchors approximation with more accurate data
Essential Math for Games

22. Multistep Methods (cont’d)
Two types of multistep methods
Explicit method
determined only from known values
Implicit method
formula includes value from next time step
Use Runga-Kutta to calculate initial values, predictor-correct for implicit
Essential Math for Games

23. Multistep Methods (Cont’d)
Adams-Bashforth 2-Step Method (explicit)
Adams-Moulton 2-Step Method (implicit)
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24. Essential Math for Games
Variable Step Size
Idea: use one level of calculation to compute value, one at a higher level to check for error
If error high, decrease step size
Not really practical because step size can be dependant on frame rate
Also expensive, not good for real-time
Essential Math for Games

25. Essential Math for Games
Which To Use?
In practice, Midpoint or Euler’s method may be enough if time step is small
At 60 fps, that’s probably the case
Having trouble w/sim exploding? Try implicit Euler or Verlet
Essential Math for Games

26. Essential Math for Games
References
Burden, Richard L. and J. Douglas Faires, Numerical Analysis, PWS Publishing Company, Boston, MA, 1993.
Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modelling, SIGGRAPH 2002.
Eberly, David, Game Physics, Morgan Kaufmann, 2003.
Essential Math for Games