Talk:MUSCL scheme

Historical perspective
Maybe, it would be better to illustrate MUSCL more in original context of upwind Godunov schemes, as it is in famous van Leer's serial of papers, and not only in semi-discrete central Godunov schemes, which is quite new concept. But I also use it now in Kurganov-Tadmor variation. --Vladimír Fuka 17:07, 7 September 2006 (UTC)
 * Possibly. I will think about it. Griffgruff 11:24, 17 May 2007 (UTC)

Proper citation
Proper style of citation in Harvard system (author, date) is not to repeat author's name in parenthesis when it is used in the sentence, for example: ... as shown by Godunov (1959) ... contrary to ... as shown in (Godunov, 1959) ... See http://en.wikipedia.org/wiki/Harvard_system. If you don't mind, I will repair it. --Vladimír Fuka 16:54, 21 October 2006 (UTC)

Rusanov contribution
The Kurganov and Tadmor scheme starts with F* reconstruction using the local maximum of the flux jacobian $$\partial F/\partial u$$. Isn't this exactly the same as the local Lax-Friedrichs method, originally described by Rusanov in J. Comp. Math. Phys. USSR, 1 (1961)? (Source: Randall J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, 2002, see pp. 232,233) --ArthurVanDam 15:25, 14 May 2007 (UTC)


 * Yes, you are correct. In fact Kurganov and Tadmor acknowledge this in their paper. I will shortly add someting to this effect. Thank you for your feedback and kind comments on my talk page. Griffgruff 11:22, 17 May 2007 (UTC)


 * Reference to Rusanov and local Lax-Friedrichs fluxes now included. Griffgruff 13:37, 18 May 2007 (UTC)

==Suggestion Best wishes —Preceding unsigned comment added by 134.134.139.70 (talk) 20:48, 26 July 2010 (UTC)
 * I think the comparisons at the end for the analytic and 2nd order and the analytic and 3rd order are nice - but a three way comparison could yield a beter metric of comparison. I think you could use an iSNR analog to indicate the difference between the two approximations.  iSNR=10*log_10(norm(analytic-2nd order)/norm(analytic-3rd order)).

Local propagation speed in the KT scheme
Currently the local propagation speed in the KT scheme is given as


 * $$ a_{i \pm \frac{1}{2} } \left( t \right) = \max \left[

\rho \left( \frac{\partial F \left( u_{i}  \left( t \right) \right)}{\partial u} \right) , \rho \left( \frac{\partial F \left( u_{i \pm 1} \left( t \right) \right)}{\partial u} \right), \right] $$

However, in Eq. (3.2) of the original paper it is defined as


 * $$ a_{i + \frac{1}{2} } \left( t \right) = \max \left[

\rho \left( \frac{\partial F \left( u^L_{i+1/2}  \left( t \right) \right)}{\partial u} \right) , \rho \left( \frac{\partial F \left( u^R_{i+1/2} \left( t \right) \right)}{\partial u} \right), \right] $$

So instead of using the cell values they use the extrapolated ones. Am I missing something here? Or is this a mistake on the page? Arno Mayrhofer (talk) 04:01, 6 February 2015 (UTC)


 * Thank you for spotting this error, which I have now corrected. Graham W. Griffiths (talk) 09:58, 6 February 2015 (UTC)

Indices in formulas
In chapter Linear reconstruction

Formula


 * $$ u^L_{i + 1/2} = u_i + 0.5 \phi \left( r_i \right) \left( u_{i} - u_{i-1} \right),

u^R_{i + 1/2} = u_{i+1} - 0.5 \phi \left( r_{i} \right) \left( u_{i+1} - u_{i} \right),$$

should be corrected to


 * $$ u^L_{i + 1/2} = u_i + 0.5 \phi \left( r_i \right) \left( u_{i+1} - u_{i} \right),

u^R_{i + 1/2} = u_{i+1} - 0.5 \phi \left( r_{i+1} \right) \left( u_{i+1} - u_{i} \right),$$

In chapter Example: 1D Euler equations

Formula


 * $$ \rho^L_{i + \frac{1}{2}} = \rho_{i}  + 0.5 \phi \left( r_{i} \right) \left( \rho_{i+1} - \rho_{i} \right),

\rho^R_{i + \frac{1}{2}} = \rho_{i+1} - 0.5 \phi \left( r_{i+1} \right) \left( \rho_{i+2} - \rho_{i+1} \right),$$

should be corrected to


 * $$ \rho^L_{i + \frac{1}{2}} = \rho_{i}  + 0.5 \phi \left( r_{i} \right) \left( \rho_{i+1} - \rho_{i} \right),

\rho^R_{i + \frac{1}{2}} = \rho_{i+1} - 0.5 \phi \left( r_{i+1} \right) \left( \rho_{i+1} - \rho_{i} \right),$$ 82.206.66.51 (talk) 16:09, 1 February 2016 (UTC)

I was having issues in my code due to the mistake mentioned just above this comment! After changing the indices in my own program, the time evolution was fixed. I second that these indices should be fixed! 216.165.95.1 (talk) 00:37, 19 February 2017 (UTC)

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Error in indexes
Hi all,

I think there is a mistake in the definition of the indexes in the chapter "Linear reconstruction". If I believe the book "Numerical Analysis Using R", the correct version should be

$$ u^L_{i + 1/2} = u_i + 0.5 \phi \left( r_i \right) \left( u_{i+1} - u_{i} \right), u^R_{i + 1/2} = u_{i+1} - 0.5 \phi \left( r_{i+1} \right) \left( u_{i+2} - u_{i+1} \right),$$

$$ u^L_{i - 1/2} = u_{i-1} + 0.5 \phi \left( r_{i-1} \right) \left( u_i - u_{i-1} \right), u^R_{i - 1/2} = u_i - 0.5 \phi \left( r_i \right) \left( u_{i+1} - u_i \right),$$

with $$ r_{i} = \frac{u_i - u_{i-1}}{u_{i+1} - u_i}.$$

With these definitions, one actually has that $$ u^R_{{i+1} - 1/2} = u^R_{i + 1/2} $$, and therefore that $$ F^*_{i+\frac{1}{2}} = F^*_{i+1-\frac{1}{2}} $$, a property that is exploited by the code GEES linked on this page, where only $$ F^*_{i+\frac{1}{2}} $$ is evaluated. This is not the case with the current version of the wikipage.