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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    m  StiffnessRunning 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    aTake two approximations Approximate at half the time step Use slope there for final approximationa    }Midpoint Method:Writing it out: Can still oscillate if h is too large&;, Runga-Kutta B          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 6M,N|            (Runga-Kutta (cont d)B           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 equations4     )    Implicit MethodsExplicit Euler methods add energy Implicit Euler removes it Use new velocity, not current E.g. Backwards Euler: Better for stiff equations$<PV        )    3Implicit MethodsJResult of backwards Euler Solution converges more slowly But it converges!@Implicit MethodsHow 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 MethodsNSolving using linear system: Resulting matrix is sparse, easy to invertOZOVerlet Integration Velocity-less scheme From molecular dynamics Uses position from previous time step Stable, but not as accurate Good for particle systems, not rigid bodyVerlet 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 data4Multistep 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`/"+L/"+L> z= 4Multistep Methods (Cont d) SAdams-Bashforth 2-Step Method (explicit) Adams-Moulton 2-Step Method (implicit)  E"Variable Step SizeIdea: 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?     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