Show that the difference methed w0 a wi 1 wi a1 ftiwi a

Show that the difference methed {w_0 = a, w_i + 1 = w_i + a_1 f(t_i,w_i) + a_2 f(t_i + a_2,w_1 + delta_2 f(t_i,w_i)), For each i = 0,1,....,N -1, cannot have local truncation error O(h^3)for any choice of constants a_1,a_2,a_2 and delta_2.

Solution

The local truncation error is given as:

T_n = y(t_n) - y(t_(n-1)) - hA(t_(n-1), y(t_(n-1)), h, f)

If the numerical method is of order n then the local truncation error is O(h^(n+1))

The given difference method is of order 2.

Hence the local truncation error will be O(h^3).