Numeric Integration Methods

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Numeric IntegrationMethods

• Marq Singer
• Red Storm Entertainment
• marqs_at_redstorm.com

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Talk Summary

• Going to talk about
• Eulers method subject to errors
• Implicit methods help, but complicated
• Verlet methods help, but velocity inaccurate
• Symplectic methods can be good for both

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Forces Encountered

• Dependant on position springs, orbits
• Dependant on velocity drag, friction
• Constant gravity, thrust
• Will consider how methods handle these

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Eulers Method

• Has problems
• Expects the derivative at the current point is a
  good estimate of the derivative on the interval
• Approximation can drift off the actual function
  adds energy to system!
• Worse farther from known values
• Especially bad when
• System oscillates (springs, orbits, pendulums)
• Time step gets large

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Eulers Method (contd)

• Example orbiting object

x
x0
x1
x2
t
x3
x4
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Stiffness

• Have similar problems with stiff equations
• Have terms with rapidly decaying values
• Larger decay stiffer equation req. smaller h
• Often seen in equations with stiff springs (hence
  the name)

x
t
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Euler

• Lousy for forces dependant on position
• Okay for forces dependant on velocity
• Bad for constant forces

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Runge-Kutta

• Idea single derivative bad estimate
• Use weighted average of derivatives across
  interval
• How error-resistant indicates order
• Midpoint method Order Two
• Usually use Runge-Kutta Order Four, or RK4

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Runge-Kutta (contd)

• RK4 better fit, good for larger time steps
• Tends to dampen energy
• Expensive requires many evaluations
• If function is known and fixed (like in physical
  simulation) can reduce it to one big formula

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Runge-Kutta

• Okay for forces dependant on position
• Okay for forces dependant on velocity
• Great for constant forces
• But expensive four evaluations of derivative

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Implicit Methods

• Explicit Euler method adds energy
• Implicit Euler dampens it
• Use new velocity, not current
• E.g. Backwards Euler
• Better for stiff equations

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Implicit Methods

• Result of backwards Euler
• Solution converges - not great
• But it doesnt diverge!

x0
x1
x2
x3
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Implicit Methods

• How to compute or ?
• Derive from formula (most accurate)
• Solve using linear system (slowest, but general)
• Compute using explicit method and plug in value
  (predictor-corrector)

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Implicit Methods

• Solving using linear system
• Resulting matrix is sparse, easy to invert

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Implicit Methods

• Example of predictor-corrector

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Backward Euler

• Okay for forces dependant on position
• Great for forces dependant on velocity
• Bad for constant forces
• But tends to converge better but not ideal

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Verlet Integration

• Velocity-less scheme
• From molecular dynamics
• Uses position from previous time step
• Very stable, but velocity estimated
• Good for particle systems, not rigid body

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Verlet Integration

• Leapfrog Verlet
• Velocity Verlet

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Verlet Integration

• Better for forces dependant on position
• Okay for forces dependant on velocity
• Okay for constant forces
• Not too bad, but still have estimated velocity
  problem

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Symplectic Euler

• Idea velocity and position are not independent
  variables
• Make use of relationship
• Run Eulers in reverse compute velocity first,
  then position
• Very stable

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Symplectic Euler

• Applied to orbit example
• (Admittedly this is a bit contrived)

x0
x1
x2
x3
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Symplectic Euler

• Good for forces dependant on position
• Okay for forces dependant on velocity
• Bad for constant forces
• But cheap and stable!

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Which To Use?

• With simple forces, standard Euler or higher
  order RK might be okay
• But constraints, springs, etc. require stability
• Recommendation Symplectic Euler
• Generally stable
• Simple to compute (just swap velocity and
  position terms)
• More complex integrators available if you need
  them -- see references

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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.

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References

• Hairer, et al, Geometric Numerical Integration
  Illustrated by the Störmer/Verlet method, Acta
  Numerica (2003), pp 1-51.
• Robert Bridson, Notes from CPSC 533d Animation
  Physics, University of BC.