Derive a third order accurate scheme

[wu_weidong]

Homework Statement

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.

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
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

 

[mighty2000]

Dear Rayne,

Thank you for sharing your attempt at deriving a third order accurate scheme for the inhomogeneous equation. Your approach using Taylor's expansion and replacing time derivatives with space derivatives is a good starting point.

However, your teacher is correct in saying that expanding up to u_ttt is not enough. This is because when using the Lax-Wendroff approach, we are essentially using a second order accurate scheme (since we are replacing the first order time derivative with a second order space derivative). Therefore, in order to achieve third order accuracy, we need to expand up to u_tttt.

Expanding up to u_tttt will result in additional terms, such as (k^4)u_tttt, which will contribute to the overall accuracy of the scheme. These terms will also need to be taken into account when using forward/central difference schemes to derive the final scheme.

I hope this helps clarify why expanding up to u_ttt is not enough for a third order accurate scheme. Keep up the good work and don't hesitate to ask for further clarification if needed.