Numeric Integration Methods Marq Singer Red Storm Entertainment marqs@redstorm.com.

Numeric IntegrationMethods

Marq SingerRed Storm Entertainment

marqs@redstorm.com

22

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

33

Forces Encountered

Dependant on position: springs, orbits Dependant on velocity: drag, friction Constant: gravity, thrust

Will consider how methods handle these

44

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

55

Euler’s Method (cont’d)

Example: orbiting object

x0 x1

x2

t

x

x4

x3

66

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

77

Euler

Lousy for forces dependant on position

Okay for forces dependant on velocity

Bad for constant forces

88

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

99

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

1010

Runge-Kutta

Okay for forces dependant on position Okay for forces dependant on velocity Great for constant forces

But expensive: four evaluations of derivative

1111

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

h

h

vvv

xxx

1212

Implicit Methods

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

x0

x1

x2x3

1313

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)

1ix 1iv

1414

Implicit Methods

Solving using linear system:

Resulting matrix is sparse, easy to invert

)()(1

))()((

)(

)(

1

11

iii

i

iiiii

iiii

iiiiii

iiii

h

h

h

h

h

xfxfIx

xxfxfx

xxfx

xxfxxx

xxx

1515

Implicit Methods

Example of predictor-corrector:

mh

h

mh

h

iiiiii

iiii

iiii

iii

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

)~(2/

/),(~

~

111

11

1

1

vxFvxFvv

vvxx

vxFvv

vxx

1616

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

1717

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 iiii h axxx 2

11 2

1818

Verlet Integration

Leapfrog Verlet

Velocity Verlet

vi1/ 2 vi hi /2a ix i1 x i hvi1/ 2

vi1 vi1/ 2 hi /2a i1

vi1/ 2 vi 1/ 2 hi a ix i1 x i hi vi1/ 2

1919

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

2020

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

2121

Symplectic Euler

Applied to orbit example

(Admittedly this is a bit contrived)

x0 x1

x2

x3

2222

Symplectic Euler

Good for forces dependant on position Okay for forces dependant on velocity Bad for constant forces

But cheap and stable!

2323

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.