Duke University
Department of Mathematics
Spring Semester, 2006
Mathematics 160S: Mathematical Numerical Analysis

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  • Homework No. 9, due Tuesday, April 11.
    Write a program that executes the following algorithms for initial value problems:
    modified Euler (author's nomenclature, p. 276-277), Heun's method (author's nomenclature, p. 274), and 4th-order Runge Kutta (RK4).
    Use all three of these methods to solve problems 5c and 5d from text page 264.
    In addition (for 5c and 5d), by reducing the step-size h by multiplying by successive factors of 1/2 (you may wish to do this more than twice), obtain approximate values for y(b) and compute approximations to the orders of convergence, the asymptotic constants, and the true value of y(b) (note that b = 2 in 5c and b = 1 in 5d). Discuss the resulting asymptotic approximations to y(b) in the context of the true value for y(b), obtained in problems 7c and 7d (p. 264).

    From the text, page 264, problems 7c and 7d, as follows:
    for this problem, for the three methods above, and also for forward Euler (as in HW8), graph the error, y(t) - (numerical approximation), on the intervals stipulated by the text in 5c and 5d (page 264), under the assumption that the numerical approximations use linear interpolation between values of wi. Also, produce graphs of error for step sizes h/2 and h/4.

    Describe and discuss the differing errors for forward Euler, modified Euler, Heun's method, and RK4, and how these errors are related to the order of convergence.

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    Last modified: 24 March 2006