The Nature of Mathematical Modelling


What is the second-order approximation error of the Heun method, which averages the slope at the beginning and the end of the interval?


For a simple harmonic oscillator y¨+y = 0, with initial conditions y(0) = 1, y˙(0) = 0, find y(t) from t = 0 to 100π. Use an Euler method and a fixed-step fourth-order Runge–Kutta method. For each method check how the average local error, and the error in the final value and slope, depend on the step size.

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Write a fourth-order Runge–Kutta adaptive stepper for the preceding problem, and check how the average step size that it finds depends on the desired local error.

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Numerically solve the differential equation found in Problem 5.3: l ¨θ + (g + z¨)sin θ = 0 . (7.34) Take the motion of the platform to be periodic, and interactively explore the dynamics of the pendulum as a function of the amplitude and frequency of the excitation.