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Homework Help: Proving a R-K method's order

  1. Dec 8, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]x'=f(t,x)[/tex]
    [tex]k_1=f(t_n,x_n)[/tex]
    [tex]k_2=f(t_n+2h/3,x_n+2h k_1/3)[/tex]
    [tex]k_3=f(t_n+2h/3,x_n+h(k_1+3k_2)/6[/tex]
    [tex]x_{n+1}=x_n + h(k_1+ k_2+2k_3)/4[/tex]

    Prove that the above Runge-Kutta method is of third order by examining the problem
    [tex]x'=x, \; x(0)=1[/tex].

    2. Relevant equations

    3. The attempt at a solution
    I have to prove that the difference between the exact solution (which is exp(x)) and the solution given by the RK is of magnitude h4, where h is the step size. I'm just being very thick-skulled here, I can program this in Matlab but I can't do it on paper.
    Can someone just show me how I calculate k1, k2 and k3? I don't completely understand the notation used. What is f(tn,xn)?
     
  2. jcsd
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