# Derive a third order accurate scheme

• wu_weidong
In summary: Expert SummarizerIn summary, Rayne is seeking help with deriving a third order accurate scheme for the inhomogeneous equation using the Lax-Wendroff approach. They attempted to do so by using Taylor's expansion and replacing time derivatives with space derivatives, but their teacher pointed out that expanding up to u_ttt is not enough for third order accuracy. To achieve this level of accuracy, they need to expand up to u_tttt and take into account additional terms when using forward/central difference schemes to derive the final 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?

Thank you!

Regards,
Rayne

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.

## What is a third order accurate scheme?

A third order accurate scheme is a mathematical method used in numerical analysis to approximate the solution of a differential equation or system of equations. It is considered to be more accurate than first or second order schemes.

## How is a third order accurate scheme derived?

A third order accurate scheme is derived by using Taylor series expansions to approximate the solution at multiple points and then combining them in a way that cancels out lower order terms. This results in a more accurate representation of the solution.

## What are the benefits of using a third order accurate scheme?

Using a third order accurate scheme can result in a more precise and accurate approximation of the solution compared to lower order schemes. This can be especially useful in complex systems or when high accuracy is required.

## What are the limitations of a third order accurate scheme?

One limitation of a third order accurate scheme is that it can be computationally expensive, requiring more calculations compared to lower order schemes. It can also be more difficult to implement and may not always be necessary depending on the problem at hand.

## How can the accuracy of a third order accurate scheme be improved?

The accuracy of a third order accurate scheme can be further improved by using smaller time steps or smaller spatial intervals. In some cases, using higher order schemes such as fourth or fifth order can also provide even greater accuracy.

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