When applying Richardson Extrapolation to Euler Method, when isn't the local truncation error of h^2 used to get (4*Euler(h/2) - Euler(h)) / 3 instead of using the global error of h to get 2*Euler(h/2) - Euler(h)?

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When applying Richardson Extrapolation to Euler Method, when isn&#039;t the local truncation error of h^2 used to get (4*Euler(h/2) - Euler(h)) / 3 instead of using the global error of h to get 2*Euler(h/2) - Euler(h)?

Szulikto · Accepted Answer

In general when you are applying Richardson extrapolation to a method   T  (  h  ), it should be an approximation of a fixed quantity   I which doesn&#039;t depend on   h. Thus for instance for the Euler method,   T  (  h  ) is the result of running the Euler method with step size h for a fixed ODE up to a fixed finite time. So the error that matters is the global error.

rhenan5v · Accepted Answer

You want to compare two approximations at   t  =  h. The error of two Euler steps at half step size is in the leading order  2  ⋅  C            (              h        2            )        2    =      1    2    ⋅  C      h    2  if the truncation error is locally   C      h    2   for a step of size   h.Thus you get, if   E  u  l  e  r  (  h  ) denotes the method application to get to a fixed time   t with step size   h,  E  u  l  e  r  (  h  )  =  y  +  C      h    2    +  O  (      h    3    )and with two steps of half the step size  E  u  l  e  r  (  h      /    2  )  =  y  +      1    2    C      h    2    +  O  (      h    3    )  .Eliminating the term   C      h    2    2  ⋅  E  u  l  e  r  (  h      /    2  )  −  E  u  l  e  r  (  h  )  =  y  +  O  (      h    3    )This is incidentally the explicit midpoint method  y  (  t  +  h  )  =  y  (  t  )  +  h  f  (  t  +            h      2        ,    y  (  t  )  +            h      2        f  (  t  ,  y  (  t  )  )  )  +  O  (      h    3    )

When applying Richardson Extrapolation to Euler Method, when isn't the local truncation error of h^2 used to get (4Euler(h/2) - Euler(h)) / 3 instead of using the global error of h to get 2Euler(h/2) - Euler(h)?

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