integration techniques

Integration Techniques

Marq Singer ([email protected])

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


The Problem

• Physical simulation with force dependant on position or velocity

• Start at x0, v0

• Only know:


The Solution

• Do an iterative solution Start at some initial value Ideally follow a step-by-step (or stepwise)

approximation of the function


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


Euler's Method

• Step across vector field of functions

• Not exact, but close

x0

x2

x1

x

t


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


Euler’s Method (cont’d)

• Example of drift

x0

x1

x2

t

x


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)


Midpoint Method

• Take two approximations

• Approximate at half the time step

• Use slope there for final approximation

h

h/2

x0.5

x0

x1

t

x


Midpoint Method

• Writing it out:

• Can still oscillate if h is too large


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


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


Implicit Methods

• Explicit Euler methods add energy

• Implicit Euler removes it

• Use new velocity, not current

• E.g. Backwards Euler:

• Better for stiff equations


Implicit Methods

• Result of backwards Euler

• Solution converges more slowly

• But it converges!

x0

x1

x2

t

x


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)


Implicit Methods

• Example of predictor-corrector:


Implicit Methods

• Solving using linear system:

• Resulting matrix is sparse, easy to invert


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


Verlet Integration

• Others: Leapfrog Verlet

Velocity Verlet


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


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


Multistep Methods (Cont’d)

• Adams-Bashforth 2-Step Method (explicit)

• Adams-Moulton 2-Step Method (implicit)

)()(32 11 iiii ffh

yyyy

)()(8)(512 111 iiiii fffh

yyyyy


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


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


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.

integration techniques

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