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Essential Math for Games: Integration Techniques in ODEs

Learn various integration techniques for solving initial value problems in ODEs. Dive into Euler's method, Midpoint method, Runge-Kutta, Implicit methods, Verlet integration, and Multistep methods. Understand their applications in physical simulations and game development.

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Essential Math for Games: Integration Techniques in ODEs

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  1. Integration Techniques Marq Singer (marq@essentialmath.com)

  2. 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 Essential Math for Games

  3. The Problem • Physical simulation with force dependant on position or velocity • Start at x0, v0 • Only know: Essential Math for Games

  4. The Solution • Do an iterative solution • Start at some initial value • Ideally follow a step-by-step (or stepwise) approximation of the function Essential Math for Games

  5. 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 Essential Math for Games

  6. Euler's Method • Step across vector field of functions • Not exact, but close x x0 x1 x2 t Essential Math for Games

  7. 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 Essential Math for Games

  8. Euler’s Method (cont’d) • Example of drift x x0 x2 t x1 Essential Math for Games

  9. 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) Essential Math for Games

  10. Midpoint Method • Take two approximations • Approximate at half the time step • Use slope there for final approximation x x0 h x1 h/2 x0.5 t Essential Math for Games

  11. Midpoint Method • Writing it out: • Can still oscillate if h is too large Essential Math for Games

  12. 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 Essential Math for Games

  13. 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 Essential Math for Games

  14. Implicit Methods • Explicit Euler methods add energy • Implicit Euler removes it • Use new velocity, not current • E.g. Backwards Euler: • Better for stiff equations Essential Math for Games

  15. Implicit Methods • Result of backwards Euler • Solution converges more slowly • But it converges! x x0 x1 x2 t Essential Math for Games

  16. 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) Essential Math for Games

  17. Implicit Methods • Example of predictor-corrector: Essential Math for Games

  18. Implicit Methods • Solving using linear system: • Resulting matrix is sparse, easy to invert Essential Math for Games

  19. 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 Essential Math for Games

  20. Verlet Integration • Others: • Leapfrog Verlet • Velocity Verlet Essential Math for Games

  21. 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 Essential Math for Games

  22. 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 Essential Math for Games

  23. Multistep Methods (Cont’d) • Adams-Bashforth 2-Step Method (explicit) • Adams-Moulton 2-Step Method (implicit) Essential Math for Games

  24. 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 Essential Math for Games

  25. 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 Essential Math for Games

  26. 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. Essential Math for Games

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