Does the Modified Euler Method always overestimate the true solution values?Here is the modified Euler Method: w 0 = α w i + 1 = w i + h 2 [ f ( t i , w i ) + f ( t i + 1 , w i + h f ( t i , w i ) ) ] , i = 0 , 1 , … , N − 1Because based on my calculations and data for a problem that I'm working on, I'm getting that it consistently overestimates the true solution...Is it supposed to do that?

Question

Does the Modified Euler Method always overestimate the true solution values?Here is the modified Euler Method:      w    0    =  α      w          i      +      1        =      w          i        +      h    2    [  f  (      t    i    ,      w    i    )  +  f  (      t          i      +      1        ,      w    i    +  h  f  (      t    i    ,      w    i    )  )  ]  ,  i  =  0  ,  1  ,  …  ,  N  −  1Because based on my calculations and data for a problem that I&#039;m working on, I&#039;m getting that it consistently overestimates the true solution...Is it supposed to do that?

Kaydence Washington · Accepted Answer

It does not always overestimate the solution for all   f, though it might do so for a particular one. Set   f  (  t  ,  y  )  =      t   and       w    0    =      t    0    =  0. We find that      w    1    =  0  +      h    2    (  0  +      h    )  =            h                        3          2                      2    .On the other hand we can compute the real solution explicitly and find that the true value in   t  =  h is       2    3        h                  3        2            .

Does the Modified Euler Method always overestimate the true solution values? Here is the modifi

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