I need help with deriving a third order accurate scheme for the inhomogeneous equation u_t + a u_x = f based on the approach used to derive the Lax-Wendroff scheme, that is, replacing the time derivatives of u by space derivatives of u.
The Attempt at a Solution
I tried to do it by using the Taylor's expansion to u(x,t+k), i.e.
u(x,t+k) = u(x,t) + ku_t(x,t) + (1/2)(k^2)u_tt(x,t) + (1/6)(k^3)u_ttt(x,t) + O(k^4)
After replacing the time derivatives of u by space derivatives of u, I
u(x,t+k) = u(x,t) - ak u_x + (ak)^2/2 u_xx - (ak)^3/6 u_xxx + kf - (1/2)(ak^2)f_x + (1/2)(k^2)f_t + (1/6)(a^2 k^3)f_xx - (ak^3)/6 f_xt + (1/6)(k^3)f_tt + O(k^4)
Then I used forward/central difference schemes on each of the term to derive the scheme.
However, my teacher said it is not enough to expand up to u_ttt, and I don't understand why. Does the u_ttt term get canceled anywhere?