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Numeric Integration Methods

This talk discusses Euler's method and its errors, implicit methods, Verlet methods, symplectic methods, and forces encountered in numerical integration, with examples of how different methods handle complex systems. The presentation compares Euler's method, Runge-Kutta, implicit methods, Verlet integration, and symplectic Euler, providing recommendations and practical considerations.

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Numeric Integration Methods

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  1. Numeric IntegrationMethods Marq Singer Red Storm Entertainment marqs@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

  3. 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 x x0 x1 x2 x3 t x4

  6. 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

  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

  9. 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

  10. 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 equations

  12. Implicit Methods • Result of backwards Euler • Solution converges - not great • But it doesn’t diverge! x0 x1 x2 x3

  13. 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)

  14. Implicit Methods • Solving using linear system: • Resulting matrix is sparse, easy to invert

  15. Implicit Methods • Example of predictor-corrector:

  16. 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

  17. 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

  18. Verlet Integration • Leapfrog Verlet • Velocity Verlet

  19. 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

  20. 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

  21. Symplectic Euler • Applied to orbit example • (Admittedly this is a bit contrived) x0 x1 x2 x3

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

  23. 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

  24. 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.

  25. 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.

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