Given a numerical scheme for the Euler equations, when do we say that it is a "genuinely multidimensional upwind" scheme ? What are the properties that a scheme has to satisfy in order to be so qualified ?

Quite frankly, I have never heard the exact term "genuinely multidimensional upwind", although I have worked extensively with schemes for the Euler equations that are both "upwind" and "multi-dimensional".

The term "upwind" is used rather loosely in CFD, but it generally does not restrict itself to the direction the wind is blowing within a flow. "Upwind" is also meant with respect to the direction acoustic information is propagated in a flow. Say in a 1-D flow, your flow speed is u, and your local speed of sound is a, then the three eigenvalues u, u-a, u+a describe the three "wind speeds", and "upwind" is meant with respect to all three.

The term "multi-dimensional" can also have a two-fold meaning: One could simply refer to a 2-D or 3-D flow, but one may also be referring to the wave character of the Euler equations itself. In essence, the Euler equations can be understood as a non-linear multi-dimensional wave equation, its dimensions stemming from the different conserved properties (mass, momentum, energy).

Let me refer you to my recently published AIAA, "Eigenvalues and Eigenvectors of the Euler Equations in General Geometries", where I talk about this stuff extensively,

www.cfd4pc.com/download/pdf/AIAA2001-2609.pdf

For example, on page 2, in the second paragraph, it says,

"The transformation matrix [A] can be interpreted as a wave speed with local and directional dependence for a nonlinear multi-dimensional wave. The multi-dimensional character is really twofold: (1) we are working in a 3-D flow field, where waves can travel in any direction; (2) there are different types of waves, all traveling at their own characteristic speeds, which are determined by the eigenvalues of the matrix [A]."

More on this topic and how to implement the eigensystem theory in different (upwind) schemes can be found in my dissertation,

www.cfd4pc.com/papers.htm

I hope this helps!

Axel

There are a couple of papers by P. L. Roe on "Genuinely multi-dimensional upwind schemes" in the late eighties or early nineties. In general, multi-dimensional upwind schemes use the concept of dimension-splitting. In the sense that it splits the wave propagation problem along the 3 axes. In reality the 5-dimensional (mass, 3*momentum, energy) problem gives rise to 5 waves that might propagate not only as linear waves but say spherical, plane or other kinds of waves...

I had read this paper long time back so do not recollect everything. In brief, there is a small body of work on "genuinely multi-dimensional upwind schemes".

chidu...

The term "genuinely two-dimensional schemes" appears in Roe's paper, "Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics". I have seen some other instances of its use but cannot recall them at present.

I agree that "upwind" is a term that is used with great liberty these days. But is it merely a matter of wrong usage or has the definition become hazy ? I am in a situation where some people object to my usage of the word "upwind" and say that it should be "upwind biased". Is this distinction really important ?

When I say multidimensional, I am not merely referring to the solution of multidimensional problems. Most of the numerical methods use a simple extension of 1-D Riemann solvers etc. when they go to multidimensions. Though they work quite well in most cases, they have not been expicitly designed to model the multidimensional physics. Their poor performance is seen in the case of flow features obliquely inclined with the grids than when they are aligned with the grid. Accurate capturing of vortices is another multidimensional issue. Some people including Roe (the reference I gave above) have developed methods which explicitly incorporate multidimensional physics and hence called "genuinely multidimensional". I am not aware of whether any substantial improvemnet has been achieved by this approach but I think many people are still working on it.

Axel, I have read your AIAA paper. Thank you for making it available freely.

" In general, multi-dimensional upwind schemes use the concept of dimension-splitting. In the sense that it splits the wave propagation problem along the 3 axes."

That is what all common "multidimensional" schemes do which is not very "physical" since coordinate directions are arbitrary. The term "genuinely multidimensional" is used when the splitting is based on multidimensional physics.

Thanks for the info, guys! I will try to get myself a copy of Roe's paper on my next trip to the library.

Work in this area began with a paper by Phil Colella written in about 1981 but not published for another five years. It was an area of significant interest in the 1990s. Phil Roe and others at the University of Michigan published a number of prominent papers. Charles Hirsch also did some excellent work in this area, in collaboration with Hermann Deconinck and others.

Characteristics are best understood as waves. In one-dimension, you have only three types of characteristics, and the "upwind" direction with respect to a given characteristic at a given point is either right or left. Easy. Physically, all information propogates from the upwind direction to the downwind direction. An upwind numerical scheme mimics this phyiscal behavior. One posting referred to "upwind bias." This refers to numerical schemes that use both upwind and downwind points for some given characteristic, but uses more points on the upwind side than on the downwind side, as opposed to "fully" upwind schemes, which use only points on the upwind side.

In two-dimensions, the situation is dramatically more complicated. You have four characteristics, and the "upwind" direction with respect to a given characteristic at a given point can be anywhere in 360 degrees. In essence, to design a "truly" or "genuninely" upwind scheme in two-dimensions, you need to understand and be able to numerically access the two-dimensional characteristics, which is tricky.

Just understanding multidimensional characteristics is very hard. Even most fluids experts don't. Most treatments do not write the characteristic equations in their most general form, but they never say so. Anything less than the most general form closes off options and flexibility which a numerical scheme might need to take advantage of.

Most numerical methods ignore the true multidimensional chacacteristics. They write the Euler equations as an x- component, a y-component, and z-component -- essentially forming dot products between the coordinate unit vectors and the true vector form of the Euler equations -- and solve for the one-dimensional characteristics of those three equations. This works great when the true multidimensional characteristics are aligned with x-, y-, and/or z-axes. It may not work so great otherwise. This is usually called dimension-by-dimension splitting and is the most common approach in practice.

The simplest "truly multidimensional" approach is to try to rotate a local x,y,z coordinate system to align it with the local characteristics. Its always hard to find the right rotation, and its possible for the chosen rotation direction to change from location to location in an unstable fashion.

"Truly multidimensional methods" were supposed to be the next big thing. Characteristics had such major success for 1D, in terms of Riemann solvers and flux vector splitting, so it seemed like it was a natural for multi-dimensions. Many, many different truly multidimensional schemes have been proposed and implemented. While its been several years since I tracked the progress in this area, at that time it seemed like no one had yet managed to devise a really practical scheme, one that actually improved on the accuracy of standard splitting approaches. Things tended to work well on one problem, and not another, or in one part of a problem and not another part.

Please excuse any inaccuracies in this description. I have written it quickly from slightly old memories. Hope it helps.

The so called genuinely multi-dimensional upwind schemes have indeed not taken off in a big as once thought. But the concept has resurfaced recently in the context of VOF methods for volume/surface tracking. For any one interested in working on advection schemes, the recent VOF literature (Ryder, Kothe, Puckett etc.) might prove useful.

As Bert pointed out, truly multi-dimensionality can perhaps be achieved only if the local axes where rotated to align with the characteristic directions. One can start working on this idea with a single non-linear wave rather than the Euler equations to begin with. However, I recently came across a scheme in which a local, higher order, multi-dimensional polynomial is built and the derivatives are estimated using this polynomial (ENO type construction). I was told one could build in some sort of upwinding while approximating the derivatives. This approach might logically be similar to FEM into which, apparently, some sort of upwinding can be built in (I can try to find out the details from a friend of mine since he was the one who mentioned it in the first place).

Dear Praveen,

Here is a review paper on the matter from 2000:

H. Deconinck, and K. Sermeus, "Status of Multidimensional Upwind Residual Distribution Schemes and Applications in Aeronautics", AIAA 2000-2328.

I think this is a very "hot" research area of CFD. The work on Multidimensional Upwind Residual Distribution Schemes started practically with the work of Phil Roe, see for instance:

P.L. Roe. Fluctuations and signals, a framework for numerical evolution problems. In Numerical Methods for Fluid Dynamics II, K.W. Morton and M.J. Baines, editors, Academic Press, pp. 219-257, 1982

P.L. Roe. Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics. Journal of Computational Physics, 63, 1986

P.L. Roe. Linear advection schemes on triangular meshes. CoA Report 8720, Cranfield Inst. Of. Tech., 1987.

P.L. Roe. Multidimensional upwinding. Motivation and concepts. In VKI Lecture Series 1994-05, Computational Fluid Dynamics, 1994.

P.L. Roe and L. M. Mesaros. Solving steady mixed conservation laws by hyperbolic/elliptic splitting. Presented at 15th International Conference on Numerical Methods in Fluid Dynamics, Monterey, 1996.

Hermann Deconinck:

H. Deconinck, Ch. Hirch and J. Peuteman. Characteristic decomposition methods for the multidimensional Euler equations. In Lecture Notes in Physics, vol. 264. Springer-Verlag, 1986.

H. Deconinck, R. Struijs and P.L. Roe. High resolution shock capturing cell vertex advection schemes for unstructured grids. Computational Fluid Dynamics VKI Lecture Series 1994-05, 1994.

H. Deconinck and G. Degrez. Multidimensional Upwind Residual Distribution Schemes and Applications. In R. Vielsmeier, F. Benkhaldoun and D. Hanel, editors, Finite Volumes for Complex Applications II Problems and Perspectives, pp. 27-40, Hermes Science Publications, Paris, 1999.

and (on the dual time stepping approach (Jameson-scheme) for unsteady Multidimensional Upwind):

D. Caraeni, L. Fuchs, ''LES Using a Parallel Multidimensional Upwind Solver''. Proceedings of the First International Conference on Computational Fluid Dynamics, ICCFD-2000, Kyoto 10-14 July, Japan, 2000.

D. Caraeni, M. Caraeni, L. Fuchs, '' A Parallel Multidimensional Upwind Algorithm for LES" AIAA 2001-2547. (It presents a new 3rd order compact multidimensional upwind scheme for unsteady Navier Stokes eqns.) If you are interested in my last paper, I can send it to you! My e-mail address is: dc@mail.vok.lth.se

Arpi Csik, a collaborator of professor Deconinck has also his own approach for the unsteady multidimensional upwind scheme (a space-time scheme), you can contact him directly: arpi@alpha14.vki.ac.be

Sincerely, Doru Caraeni

Thank you for the references. They are very helpful. In fact the first paper can be found at http://www.vki.ac.be/~sermeus/cfd/index.html