Hi all,

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.

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

got

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?

Please advise.

Thank you!

Regards,

Rayne