The question is to prove that error order of backward euler method is o ( h ) We know that in backward euler method f i ′ = f i − f i − 1 h By using taylor seris we can get f ( x ) = f ( x i ) + ( x − x i ) f ′ ( x i ) + ( x − x i ) 2 2 ! f ″ ( x i ) + . . . .By putting x = x i − 1 i get : f i − f i − 1 h + f ′ i = h 2 ! f i ″ + h 2 3 ! f i ‴ + . . .Which is not what i want to get Cause the error is f i − f i − 1 h − f i ′ Any help ? I could prove that o ( h ) is the order of error of forward Euler method by using x = x i + 1 But for the backward method it seems it doesn’t work

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The question is to prove that error order of backward euler method is   o  (  h  ) We know that in backward euler method      f    i    ′    =                    f        i            −              f                  i          −          1                      h  By using taylor seris we can get  f  (  x  )  =  f  (      x    i    )  +  (  x  −      x    i    )      f    ′    (      x    i    )  +            (      x      −              x        i                    )        2                    2      !            f    ″    (      x    i    )  +  .  .  .  .By putting   x  =      x          i      −      1       i get :                    f        i            −              f                  i          −          1                      h    +      f    ′    i  =      h          2      !            f    i    ″    +            h      2              3      !            f    i    ‴    +  .  .  .Which is not what i want to get Cause the error is                    f        i            −              f                  i          −          1                      h    −      f    i    ′  Any help ? I could prove that   o  (  h  ) is the order of error of forward Euler method by using   x  =      x          i      +      1       But for the backward method it seems it doesn’t work

Teagan Zamora · Accepted Answer

You also need to take into account that   x  −      x    i   at   x  =      x          i      −      1       has the value   −  h.Check also the other signs, the Taylor terms should be alternating.

The question is to prove that error order of backward euler method is o(h) We know that in backward euler method f′_i=(f_i−f_(i−1))/h

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