gistfile1.txt

Module[{g, d, \[Eta], m = 4, order = 3},
 \[Eta] = u s/h;
 (*d[x_]:=0;*)
 (*d[x_]:=(f[0,x+h/2]-f[0,x-h/2])/h;*)
 
 d[x_] := (2 - \[Eta])/3*(f[0, x + h/2] - f[0, x - h/2])/h + (1 + \[Eta])/3*(f[0, x - h/2] - f[0, x - 3 h/2])/h;
 g[x_] := f[0, x - h/2] + 1/2 (1 - \[Eta]) h d[x];
 Series[
     (f[s, h/2] - f[0, h/2])/s + u (g[h] - g[0])/h
     , {s, 0, m}, {h, 0, m}] // Normal // Expand
  // ReplaceRepeated[#, {
     Derivative[j_, k_][f][0, 0] :> Subscript[\[Alpha], j, k],
     Subscript[\[Alpha], 0, k_] :> Subscript[\[Alpha], k],
     Subscript[\[Alpha], j_, 
      k_] :> -u Subscript[\[Alpha], j - 1, k + 1]
     }] &
 ]
Collect[% /. s -> h \[Eta]/u, h, Simplify]