A Russell
06-22-2012, 12:23 PM
Hi,
I may have found an error in the text of the 3rd edition. Can anyone please confirm this for me?
I refer to equation (20.1.37), which describes the Lax step of the Lax-Wendroff method for solution of flux conservative PDEs. In the third edition and electronic version (on 22 June) this reads:
\begin{eqnarray}
u_{j+1/2}^{n+1/2} &=& \frac{1}{2}\left( u_j^n + u_{j+1}^n \right) - \frac{\Delta t}{\Delta x}\left( F_{j+1}^n - F_j^n \right)
\end{eqnarray}
I believe the $ \Delta t $ on the right-hand side should be replaced by $ \Delta t / 2 $, so that the step finishes at $ t+\Delta t / 2 $ as indicated on the left-hand side. The replacement equation would be:
\begin{eqnarray}
u_{j+1/2}^{n+1/2} &=& \frac{1}{2}\left( u_j^n + u_{j+1}^n \right) - \frac{\Delta t}{2 \Delta x}\left( F_{j+1}^n - F_j^n \right)
\end{eqnarray}
which appears in the second edition. The proposed form is also consistent with other resources.
As written at present, I think the scheme still solves a set of equations consistently, but the truncation error is second order, rather than third order. It also makes the stability criteria more stringent compared to the expected result that the Courant condition be satisfied for all waves. This means it would be possible to use the printed scheme (perhaps unawares) but at the cost of lower accuracy, longer run times and slower convergence to the true solution with increasing gridpoints and timesteps.
So what are people's thoughts? Have I missed something or is this a genuine typo?
Alex Russell
I may have found an error in the text of the 3rd edition. Can anyone please confirm this for me?
I refer to equation (20.1.37), which describes the Lax step of the Lax-Wendroff method for solution of flux conservative PDEs. In the third edition and electronic version (on 22 June) this reads:
\begin{eqnarray}
u_{j+1/2}^{n+1/2} &=& \frac{1}{2}\left( u_j^n + u_{j+1}^n \right) - \frac{\Delta t}{\Delta x}\left( F_{j+1}^n - F_j^n \right)
\end{eqnarray}
I believe the $ \Delta t $ on the right-hand side should be replaced by $ \Delta t / 2 $, so that the step finishes at $ t+\Delta t / 2 $ as indicated on the left-hand side. The replacement equation would be:
\begin{eqnarray}
u_{j+1/2}^{n+1/2} &=& \frac{1}{2}\left( u_j^n + u_{j+1}^n \right) - \frac{\Delta t}{2 \Delta x}\left( F_{j+1}^n - F_j^n \right)
\end{eqnarray}
which appears in the second edition. The proposed form is also consistent with other resources.
As written at present, I think the scheme still solves a set of equations consistently, but the truncation error is second order, rather than third order. It also makes the stability criteria more stringent compared to the expected result that the Courant condition be satisfied for all waves. This means it would be possible to use the printed scheme (perhaps unawares) but at the cost of lower accuracy, longer run times and slower convergence to the true solution with increasing gridpoints and timesteps.
So what are people's thoughts? Have I missed something or is this a genuine typo?
Alex Russell