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    Improved Euler method with Butcher Tableau

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    Use the order conditions to prove that the improved Euler method with Butcher Tableau

    0 |
    1 | 1
    ---+-------------
    | 1/2 1/2

    is second order but not third order.

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    https://brainmass.com/math/ordinary-differential-equations/improved-euler-method-with-butcher-tableau-433173

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    Problem Statement:

    Use the order conditions to prove that the improved Euler method with Butcher Tableau

    is second order but not third order.
    Order Conditions:
    First, let us write out what the order conditions are for a general r-stage Runge-Kutta method. Say we have the following Butcher tableau for our method:
    c_1 a_11 〖... a〗_1r

    ⋮ ⋮ ⋮
    c_r a_r1 ...〖 a〗_rr
    b_1 ...b_r

    To guarantee at least first order accuracy, and in addition ...

    Solution Summary

    The expert examines the improved Euler method Butcher Tableau is depicted.

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