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autofly April 23, 2003 04:38

MUSCL interpolation scheme is only suitable for uniform grid.if the grid space vary greatly,it leads to too much errors. is it true? if not, please give a wise interpration and give some advice about MUSCL's use in 3D

Mike April 23, 2003 08:58

MUSCLE is just an extrapolation of flow variables to the cell faces. After extrapolation, fluxes are computed at the cell faces. So I think it is not meaningful to relate solution accuracy of MUSCLE to grid spacing.


Mike April 23, 2003 10:48

So you can obviously use MUSCLE scheme in your code for 2d or 3d. Also you will do better if you use limiter with high order accuracy (like Osher-Chakr. limiter up to 3rd order acc.).


Jean-François Simon April 24, 2003 12:02

A "good" MUSCL extrapolation is second order accurate on a regular mesh. But, it is less accuarate (in between first and second order) on stretched mesh. So I think that the accuracy of MUSCL is related to the grid spacing.


Mike April 24, 2003 14:13

Jean, Using the third order accurate scheme, I experienced very good solutions for the viscous flows (ROE, k-e turb model) with high aspect ratio cells. What is the aspect ratio of your stretched cells? and what is the amount of sudden expansion in your cell sizing? Mike

Jean-François Simon April 25, 2003 07:06

Hi Mike,

I think I have misunderstand something. Could you please enlight me on the following points ?

For me, when a scheme is third order accurate, it means that the error is of the order of (\Delta x)^4 and that it can exactly represent a cubic variation. Is it right ? Is your scheme second order for me ?

With a MUSL interpolation with three points you can have a second order accurate interpolation with parameter \eta = 1/3. But, in 2D, if the mesh is skewed, the accuracy diminish because MUSCL is not really bidimensional but two times monodimensional. Am I wrong ?

Another question on scheme accuracy : even with an exact (irrealistic) interpolation (F_i+1/2 and F_i-1/2 exact), you can only have second order accuaracy because you discretize your derivative (dF/dx)_i by (F_i+1/2 - F_i-1/2)/( \Delta x) + O( \Delta x)^2. Is it right ? This is for finite difference but in finite volume, I think that there is something about compensation of errors but I can't rememenber what. Could you explain me, please ?

Thank you for your time


Mike April 27, 2003 13:27

Jean, If I can have your email, I would like to send you a pdf file which can enlighten you for the Muscl with FINITE VOLUME method. Thanks...

Andy April 27, 2003 13:42

The MUSCL scheme is a one-parameter family of schemes, where the choice of the parameter gives different schemes; Van Leer originally developed it. For instance if you choose kappa, the parameter to be 0.5, then you get the QUICK scheme. Whatever you choose the formula has been derived for a structured grid. It assumes equal spacing between grid points, and requires local co-linearity of points, i.e. they must be in a straight line. This can only be achieved with an orthogonal structured grid. There are some papers that for instance introduce a curvature factor, to take account of non co-linearity, and other papers that introduce a stretching factor, to take account of non local grid spacing but essentially they are there to adapt something that is designed for structured grids to be made usable on unstructured grids.

Since Barth of NASA introduce linear reconstruction and reconstruction in general (1989 ish) MUSCL has not used in proper academic codes on unstructured grids. These have no requirement of grid spacing and do not require a local co-linearity of points; importantly they reduce the dependence of the convective differencing on grid spacing and quality. The most important point is that in a MUSCL framework a multi-dimensional limiter cannot be used and only simple min-mod limiters can be used, which clips the primitive variables non-physically and are not differentiable. Using a multi-dimensional limiter with linear reconstruction allows the gradients to be limited physically, thereby allowing a more correct solution and a limiter that is differentiable.

The problem is that most people doing cold flow do not realize this, and it is in fact excellently highlighted in combustion LES. I would recommend switching if you have the chance and time.


Jean-François Simon April 28, 2003 03:16

Thank you Mike,


mihirmakwana6 July 3, 2015 05:05


Originally Posted by Mike
Jean, If I can have your email, I would like to send you a pdf file which can enlighten you for the Muscl with FINITE VOLUME method. Thanks...

Hello sir.

I am trying to implement MUSCL scheme for 1d advection problem using FVM.

Can you please mail me the pdf if possible to

Tony12 August 6, 2015 14:40

(replying to a 12 year old thread for the sake of future visitors to this question)

People use the term "MUSCL scheme" ambiguously to mean either using cell reconstruction for higher-order spacial accuracy, using (van Leer's) slope/flux limiters, and/or both. Be careful to specify which you are referring to.

Regarding flux/slope limiters' grid spacing, grid dimension, and cell skewness, read Berger and Aftosmis "Analysis of Slope Limiters on Irregular Grids". They probably will cover your question.


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