Numeric IntegrationMethods

Jim Van VerthRed Storm Entertainment

jimvv@redstorm.com

2

Talk Summary

Going to talk about: Euler’s 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

4

Euler’s 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

5

Euler’s Method (cont’d)

Example: orbiting object

x0 x1

x2

t

x

x4

x3

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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)

t

x

7

Euler

Lousy for forces dependant on position

Okay for forces dependant on velocity

Bad for constant forces

8

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 (cont’d)

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

11

Implicit Methods

Explicit Euler method adds energy Implicit Euler dampens it Use new velocity, not current E.g. Backwards Euler:

Better for stiff equations11

11

++

++

+=+=

iii

iii

hhvvv

xxx

&

&

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Implicit Methods

Result of backwards Euler Solution converges - not great But it doesn’t diverge!

x0

x1

x2x3

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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)

1+ix& 1+iv&

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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:

mh

h

mh

h

iiiiii

iiii

iiii

iii

/)),()~,~((2/

)~(2/

/),(~

~

111

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1

1

vxFvxFvv

vvxx

vxFvv

vxx

+⋅+=+⋅+=

⋅+=⋅+=

+++

++

+

+

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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

bodyiiii h axxx ⋅+−= −+

211 2

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Verlet Integration

Leapfrog Verlet

Velocity Verlet

€

vi+1/ 2 = vi + hi /2 ⋅a ix i+1 = x i + h ⋅vi+1/ 2

vi+1 = vi+1/ 2 + hi /2 ⋅a i+1

€

vi+1/ 2 = vi−1/ 2 + hi ⋅a ix i+1 = x i + hi ⋅vi+1/ 2

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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 Euler’s 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.