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Midpoint Euler method, second order system

  1. Jun 12, 2013 #1

    I need to plot this simple system:

    x'' = -x

    using midpoint Euler.

    u1 = -x , u2 = -x'

    u1' = u2
    u2' = -x

    u1(n+1) = u1(n) + h*?
    u2(n+1) = u2(n) + h*f((1/2)*(u1(n) + u1(n+1))

    We dont know u1(n+1). I tried approximating it with u1(n+1) = u1(n) + h*u2(n)

    u2(1+i) = u2(i) + h * ((-1/2) * (u1(n) + u1(n+1))

    Then we have u2(i+1) and u2(i). A midpoint value is (u2(1+i) - u(i))/2

    u1(q+i) = u1(i) + h*midpoint

    When i plot this the result is some awful diverging line, not an oscillating function. What is wrong?
  2. jcsd
  3. Jun 12, 2013 #2
    If you are using mid-point euler like this, then you need to move all the n+1 terms to the left sides of the two equations, and solve for u1(n+1) and u2(n+1) treating the equations as two simultaneous linear algebraic equations in two unknowns.
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