Adaptive Runge-Kutta

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• Adaptive Runge-Kutta
• addresses the problem of functions that change
  rapidly at a point

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• Would like to use small size steps in the area of
  rapid change - normal size steps in area of
  normal change
• Two approaches behind adaptive step size
• look at difference between predictions with
  different step sizes but same order RK
• look at difference between predictions with
  different order RK

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Step-halving or adpartive Runge-Kutta let y1 be
single-step prediction let y2 be prediction using
two half steps
The correction is
fifth order accurate
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Example
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Integrate y from x0 to 2 using h2, and improve
using adaptive RK
Complete step results are
Half step results are
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The correction is
The corrected value is
Compare to true value y(2)2.524369
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Runge-Kutta-Fehlberg Uses two different RK
predictions of different order Special choice of
methods lets you use results from 4th order in
5th order RK - then combine them
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Fourth order RK
Fifth order RK
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Formula for ks
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Example
Use h2, and the RKF method
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The results are RK42.542811 RK52.554121 and
EaRK5-RK42.554121-2.5428110.01131 Now adjust
stepsize
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If Ea is too small, increase step size If Ea is
too large, decrease step size
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Stiffness stiff equation involves rapidly
changing parts and slowly changing parts
Solution is
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(No Transcript)
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Look at homogeneous part of equation
In general
Explicit Eulers method
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Look at what happens to y over long time -
stability
If then y goes to infinity So for explicit
method to work - small h
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Need to use implicit methods, rather than
explicit Implicit form of the Euler method
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Implicit Euler is always stable - as i
increases y goes to 0
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Example
Explict solution since a is 2000, let h0.0001
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Stability limit is h0.0005 Try h0.0007
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Try h0.001
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h0.002
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Implicit approach
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h0.002