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S. Denisov MNTF, Uni Augsburg

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An example: integration of nonlinear oscillator Equation of motion

¨

x=−γx˙−x−x3+asin(ωt) Two variables: coordinate x and velocity υ= ˙x.

Initial conditions: This is the state of the system at the initial instant of timet = 0,x(0) = 0 and υ(0) = 0.

Task: to propagate the system over the time t = 10T, where T = 2π/ω.

Overall error: E(N) =p

(x(sN)−x(N))2+ (υ(sN)−υ(N))2, s = 4.

Algorithms: Euler’s scheme and the 4th-order Runge-Kutta method(will discuss them latter on).

Number of stepsN is the number of integration steps per period T. It sets the time-step h=T/N.

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An example: integration of nonlinear oscillator

Performances of the two algorithms

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Euler method

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RK4 method (N= 500)

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