This is an open question regarding why solutions 'blow up' at seemingly even reasonably low velocities for low-speed, viscous, incompressible flow fields.

I have noted that there are certain combinations of parameters at very low Reynold's number where solution convergence is simply impossible without the addition of solution stabilisation eg. upwinding etc.

I have looked through a number of books & references to critical cell Peclet number & critical cell Reynold's number are mentioned.

It appears to me that current theory is split into two areas , 1. Low-speed, viscous, incompressible; 2. High-speed, inviscid, compressible.

I have not been able to locate good theory for the region between these two extremes & observe that most approaches seem to use solution-stabilisation techniques eg. upwinding, to cope with the intermediate region.

I have an idea about why the solutions 'blow-up' based on new mathematical theory I have been developing, but would like to test this against existing knowledge & theory. This new theory predicts the use of very small element dimensions to overcome the 'singularity point'.

I would value feedback from the learned members of this board.

diaw...

You may have to define your problem a little clearer. There will not be any simple and general answer to the question why numerical solutions blow up, because of a quite complex interaction of possible causes, such as numerical instability of the main algorithm, ill-posedness of the physical/numerical problem, instability of the boundary conditions (maybe related to ill-posedness), unresolved physical instability... and so on. I think that you a focusing on the stability of your numerical algorithm, but I wasn't quite clear from your post. Stability analysis is not a new field, you are surely familiar with the linear Von-Neumann analysis, for example. If you're interested in nonlinear stability analysis, I am not sure there is a lot you can do for the general case, but I would say that the stability issues you mentioned in your post are understood within the framework of linear stability. In order to get feedback on your new idea, you probably need to be more specific.

Thanks Mani... always a good comment...

Try this little experiment:

1. Create rectangular domain - dx=4; dy=1 2. Discretise into coarse structured elements - say 0.5x0.5; 3. Set upper & lower faces to u,v=0 4. Apply input velocity at central node on left edge (the inlet). 5. Without using flow-field stabilisation, begin with velocity u=1 m/s & gradually raise the velocity until a reasonable Reynolds is reached. 6. Fluid properties, density = 1, viscosity = 1.

Observe the total-velocity & pressure fringe plots & tell what you see. At what velocity does the solution blow up - !!!'without flow stabilisation'!!!

Interesting results on my end in FEM... at low Reynolds numbers...

This experiment was constructed in order to challenge a new 'singularity theory' I have been dabbling with on the sidelines for a few months... I can show the mathematical reasoning rather clearly.

diaw...

What do you exactly mean by 'flow stabilization'. When you apply no stabilization, what scheme do you use, i.e. what is the base scheme that you consider 'free of stabilization'. The term stabilization is a bit arbitrary, so please describe your base scheme and type of stabilization. Is the choice of your base scheme unique? Would everyone else choose the same base scheme and call it non-stabilized? I think that's up to interpretation. Is your problem well-posed? (If it isn't, your findings/method may be difficult to generalize to more relevant well-posed problems.)

Thanks again Mani... good points.

What I meant by 'no flow stabilisation' is no use of upwind, or other convection/diffusion-modification methods - just the basic discretisation scheme of the governing pde's eg. basic application of MWR-Galerkin to continuity, x-momentum, y-momentum (energy off for now).

In the literature we read about a 'critical cell Reynolds number' of 2 which places an upper bound on the maximum length scale of the calculation cell.

My original test case as explained picks up the singularity covered by this 'critical cell Re', altough at slightly higer value (an indication of the diffusive nature of the numeric scheme) in direct MWR-Galerkin techniques, but shows absolutely nothing in Split-schemes (FEM) & Finite-Volume Method (Flo++).

My premise is that one fundamental singularity lies at the point defined by this 'critical Re' & that this tends to govern much of the low-speed, incompressible flow simulations we perform - wheter we know it, or not.

In fact, the system of N-S equations, is full of 'singularities' - much like a case of Swiss cheese.

Many of these 'split schemes' & pure upwinding are, by their unconscious design, able to move the dominant singularity/ies to infinity, thus allowing the user freedom from the efects of these singularity/ies.

For the energy eqn we talk about a Peclet numer of 2 as governing cell scale - why? Anwser: another 'singularity'.

Please email me as I do not have your direct email... we can then go into deeper details.

diaw...

I think you should be careful about your terminology here:-

"In fact, the system of N-S equations, is full of 'singularities' - much like a case of Swiss cheese"

Since the only singularities I know of in the Navier-Stokes equations lie in the complex Reynolds number plane - it is believed that non of these singularites lie/cross the positive real line (although proof of this is a million dollar question).

Thanks Tom,

Let me answer your excellent comment with a question by perhaps putting it this way (in my typical rough engineering talk)...

... it really depends on what lens we are looking through.

For instance, control engineers have for a long time been able to identify pertinent points of interest via the Laplace Transform (for 1D).

Let me add a few questions:

Is the reason why typical mathematics approaches cannot find the singularities is that the 'differential' 'transformation' process simply moves all the singularities to the origin as we decrease delta'x' to zero.

What if we were to develop another transform that could expose these items for us?

What is the 'cell critical Reynolds number'? Why is it exactly 2?

What is the 'cell critical Peclet number'? Is is 1 or 2? Why do convection-diffusion problems blow up if we exceed these Pe,crit & Re,crit values?

From where do these numbers originate?

-------- By way of background, I did some post-graduate research in Advanced Control Methods many years ago, & have only recently returned into the academic field after some 15 years in the Engineering Consulting field. So in many ways, I have tended to approach the numerics problems in a very unorthodox style. I have found it helpful to explain some of the hurdles we come across in the CFD field.

I value the open debate...

diaw...

Some possible answers to your questions:-

(1) "Is the reason why typical mathematics approaches cannot find the singularities is that the 'differential' 'transformation' process simply moves all the singularities to the origin as we decrease delta'x' to zero."

I suspect that the singularities shoot off into the complex plane as the grid spacing is refined. The appearance of them on the real line is an artefact of the discretization. A good example of a discrete problem behaving very differently to the continuous one is to discretize the logistic equation using a forward Euler scheme. The resulting discrete equation possesses a period doubling cascade and chaos as the free parameter (timestep) is varied while the continous problem has no such behaviour. In the limit of the timestep going to zero the discrete problem approaches the continuous soultion but outside this range there is no reason for them to share the same behaviour.

(2) "What if we were to develop another transform that could expose these items for us?"

Quite possibly - the Z transform is often used for discrete (linear) problems.

You may be interested in a paper by Stephen Cowley "Computer-extension and analytic continuation of Blasius expansion for impulse flow past a circular-cylinder" Journal of Fluid Mechanics vol 135. This paper considers the motion of singularities in the complex plane for the solution of the boundary-layer equations - when a singularity encounters the real-line then the solution of the continuous problem fails to exist.

(3) "What is the 'cell critical Reynolds number'? Why is it exactly 2?"

This is nothing more than a resolution problem - if a sharp gradient occurs then the grid Reynolds number has to be sufficiently small that the gradient can be smoothed out. The 2 is due to the fact that the diffusion operator spans two grid lengths. When you upwind you effectively add in more viscosity which changes the grid Reynolds number.

The Peclet number condition is the same as the Reynolds number condition.

(4) "Why do convection-diffusion problems blow up if we exceed these Pe,crit & Re,crit values?"

Basically it's the same reason for the stability constraint - a sharp gradient is forming through advection (think of shock formation in the 1D inviscid Burger equation) and the diffusion is not strong enough (grid not fine enough) to smooth out (resolve) the sharp gradient - in the case of the inviscid Burger's equation u_x -> infinity.

Tom.

Hi Tom,

Thank you so much for providing such deep, insightful answers to my dilemma. You have provided me much additional food for thought.

Q: diaw: (1) "Is the reason why typical mathematics approaches cannot find the singularities is that the 'differential' 'transformation' process simply moves all the singularities to the origin as we decrease delta'x' to zero."

A: Tom: I suspect that the singularities shoot off into the complex plane as the grid spacing is refined. The appearance of them on the real line is an artefact of the discretization. A good example of a discrete problem behaving very differently to the continuous one is to discretize the logistic equation using a forward Euler scheme. The resulting discrete equation possesses a period doubling cascade and chaos as the free parameter (timestep) is varied while the continous problem has no such behaviour. In the limit of the timestep going to zero the discrete problem approaches the continuous soultion but outside this range there is no reason for them to share the same behaviour.

diaw's reply: I would tend to agree with the singularities being squeezed somewhere as dx is reduced to zero - perhaps they then squeeze onto the complex plan to be reflected there - nice point.

I would also strongly agree on the nature of the discretisation scheme modifying the nature of an underlying physical phenomenon - in numerical magnitude, but would believe that if the discretisation scheme is adequate, then forcing the discrete dimension to zero should display the original continuous equation, or at least, very close to it.

Q: diaw: (2) "What if we were to develop another transform that could expose these items for us?"

A: Tom: Quite possibly - the Z transform is often used for discrete (linear) problems.

You may be interested in a paper by Stephen Cowley "Computer-extension and analytic continuation of Blasius expansion for impulse flow past a circular-cylinder" Journal of Fluid Mechanics vol 135. This paper considers the motion of singularities in the complex plane for the solution of the boundary-layer equations - when a singularity encounters the real-line then the solution of the continuous problem fails to exist.

diaw's reply: The z-transform is useful for more than one space dimension?

Thank you so much for the paper reference - I will get hold of it. I seem to think that von Drommelin worked/works in this area.

Q: diaw: (3) "What is the 'cell critical Reynolds number'? Why is it exactly 2?"

A: Tom: This is nothing more than a resolution problem - if a sharp gradient occurs then the grid Reynolds number has to be sufficiently small that the gradient can be smoothed out. The 2 is due to the fact that the diffusion operator spans two grid lengths. When you upwind you effectively add in more viscosity which changes the grid Reynolds number.

The Peclet number condition is the same as the Reynolds number condition.

diaw's reply: I wholeheartedly agree that the addition of additional viscosity modifies the problem. I believe that it effectively sends the momentum & energy singularities to infinity on the real plane. The additiona of artificial viscosity causes the convection term to diminish in magnitude compared to the diffusive term & thus stabilises the solution. In fact, in 1D, application of full-upwinding (FEM) will merrily send the real singularity off to meet infinity...

Q: diaw: (4) "Why do convection-diffusion problems blow up if we exceed these Pe,crit & Re,crit values?"

A: Tom: Basically it's the same reason for the stability constraint - a sharp gradient is forming through advection (think of shock formation in the 1D inviscid Burger equation) and the diffusion is not strong enough (grid not fine enough) to smooth out (resolve) the sharp gradient - in the case of the inviscid Burger's equation u_x -> infinity.

diaw's reply: This is a very reasonable argument.

------ So far, Tom, to perhaps paraphrase things somewhat simplistically, it looks like you have your singularities on the complex plane & I have mine in the real plane. Our dividing fence is action of the 'differential' 'transform'...

Tom, thanks so very much for your excellent replies & debate. I would be honoured if you would like to discuss more privately on my email at <des@adtherm.com>

diaw...

hi diaw, i saw your interesting and engaging discussion and thoug i am a new enterant to cfd, i think i gripped some basic concepts by nonw. I dont know why the pecelt number gets difficult to u, the number setting to 2 has very physicl argument. i dont think singularity is as such involved here unless we are talking about very comples flow. the simple reason we use upwinding scheme, is that when we talk of advection the volume or mass flow rate is not the average of the value in the consecutive nodes,like temperature, etc the flow effect is only for its immediate vicinity of the node and in the upwind direction, this have long been demonstrated by patankar, in 1979, he used the argument similar to a decay function of the volume flow rate efect,

my explanation is very layman but the basic concept is there, just the same as lumped capacitance method model of heat generating body, a decaying function case, go and clear yourself by reading patankar s book

Hi taw,

Thanks very much for your addition to the debate (contents re-listed at bottom of this posting).

I consider Patankar to be a truly excellent reference source - the teachers' teacher through-&-through.

As I understand Patankar's approach is that he considers the unknowns eg. u, v, w, T, P to be constant over each cell volume. Versteeg extends things somewhat into simpler layman's language. Versteeg seems to argue that the upwinding scheme was initially introduced to provide solution stability for velocities which would exceed the critical Reynolds & Peclet numbers. Once the 'false diffusion' issue became a problem, then all sort of alternative schemes have been proposed.

I have used Patankar's exponential decay function in deriving alternative element interpolation functions (FEM). Sadly, they too succumbed to critical Peclet & Reynolds.

By strange coincidence I find that in 1D, the x-momentum equation provides a singularity at exactly the position embodied in the critical Reynolds number, and for energy equation at the critical Peclet. This would seem to go along with Tom's argument about numeric reasons for the apparent existence of these singularities. This occurs in finite diffeence, finite volume & finite element techniques... all at exactly the same point.

Of further interest is that my mathematical reasoning allows extension into multiple spatial dimensions, thus suddenly exposing coupled effects between the relevant pde's... Hence my earlier comment about the system of Navier-Stokes equations being somewhat like swiss-cheese. I have extended reasonably far into the 2D range, & once this is reasonably clear, hope to move further into the 3D nether-world.

What if real singularities do exist, or that perhaps we are really striking the border of the suitable range for say the viscous flow assumption? How would we measure such 'real singularities' since all measurment apparatus has finite bandwidth? This has been an intriguing question for me.

More interesting still is if one then looks at Bejan's approach to the Scaling Laws...

diaw...

----------------- taw's comment: I dont know why the pecelt number gets difficult to u, the number setting to 2 has very physicl argument. i dont think singularity is as such involved here unless we are talking about very comples flow. the simple reason we use upwinding scheme, is that when we talk of advection the volume or mass flow rate is not the average of the value in the consecutive nodes,like temperature, etc the flow effect is only for its immediate vicinity of the node and in the upwind direction, this have long been demonstrated by patankar, in 1979, he used the argument similar to a decay function of the volume flow rate efect,

my explanation is very layman but the basic concept is there, just the same as lumped capacitance method model of heat generating body, a decaying function case, go and clear yourself by reading patankar s book

Dear diaw, the thing u are raising is very intersting indeed, but my answer is also the basics. i am also intrigued with scaling issues rigt now with my work and i would really further look into the area from some of your perspective. but the mre complicated the discussion becomes to me b/c i am at the moment less versed with fem right now. but thankyou and i will like to follow the outcome of the discussion taw

Hi again taw,

I personally find the whole scaling issue to be fascinating. Imagine if singularities did exist in the real plane - how would that alter our perspective of scaling & simulation?

For instance, is there a correlation between real-scales & simulation scales? What is the origin of turbulence? Could it be that we don't have measuring equipment sensitive-enough to measure its very origins? Could the turbulent issue be purely a linear one originating from the instant of flow, but too small to measure until we reach the 'right measurement scale'? Nature is indeed a fascinating beast?

An alternative is that the Navier-Stokes equations have not got it quite right...

diaw...

------------ taw's reply: i am also intrigued with scaling issues rigt now with my work and i would really further look into the area from some of your perspective. but the mre complicated the discussion becomes to me b/c i am at the moment less versed with fem right now.

I have a basic question that is kind of answered in your discussion with Tom, but maybe you can comment to clarify.

Given that the stability is related to a grid Reynolds number, which is really a parameter that arises in the discretized problem and seems to have little to do with the continuous N-S equations, wouldn't you assume that any singularities you might encounter are really singularities of the discretized equations (not singularities of the N-S equations)?

If you agree, this goes back to my original point. The issue here is really the stability of your numerical scheme which more or less adequately approximates the continuous equations. The necessity of stabilization methods such as artificial dissipation, upwinding, a.s.o. is really a penalty you have to pay for discretization, because the discretized equations behave quite differently from the continuous ones (taw's comment on the physical reason's for upwinding are quite appropriate). So it's really about discretization and not about singularities of the N-S equations. Maybe this is what you meant, and I just misunderstood you.

Hi Mani,

Good to have your input again.

I see Tom's point in that the 'differential transform' basically squeezes the singularities either to zero, or to the complex plane, as we decrease the dx dimension to zero.

I had not thought about the option of the singularities - if they do exist - squeezing to any place other than zero. This has added some additional food-for-thought.

Basically, after many recent discussions with a Control Theory guru I studied under many moons back, I began trying to originally work backwards from a discretised equation, gradually decreasing the dimension 'dx', 'in the limit' to zero to understand whether the problem lay in the method of discretisation, or possibly in the underlying physical governing equation. This began to get complicated & I decided to move in a different direction.

We have to really understand that we have governing equations in a 'differentially transformed space' in which we do not compute. We have no way of ever obtaining 'in the limit to zero' on a computer - we will always have a finite computation 'distance'.

When we begin understanding things using a simple transformation, the underlying structure of any pde, or system of pde's emerges. Laplace does this for 1D linear systems - multi-dimensional non-linear systems require some different mental acrobatics, but, with the right tools, they can be made manageable.

When the underlying structure becomes evident, then the connections between the pieces also become more clear.

I am working on tools which allow us to easily investigate the effect of altering one, or more variable simultaneously, to investigate their effect on other inter-connected variables.

I will certainly update this board as more progress is made, & things are nearing publication.

---------

The effect of adding additional 'artificial viscosity' in simulations appears to merely to move the perceived singularities & stumbling-blocks to simulation, to infinity.

I do see Tom's viewpoint about trying to draw conclusions from within a discretised scheme as one would then be subject to the scheme itself. I have not done that - I have performed the transformations directly on the governing equations themselves.

The major hurdle so far has been understanding how to tie all the assorted variables into a cohesive, manageable system. This phase was passed a short while back.

----------

I must say that as I have tested my findings using a number of simulation tools, I do seem to be landing in the ballpark of the predicted 'singularities', with the main deviation potentially attributable to numeric diffusion built into the numeric scheme.

I am currently running tests on flow over a cylinder, close to the turbulence point - with absolutely no convection stabilisation whatsoever. This has been an enlightening exercise.

----------

You see, what we are really down to - at least as I see things - is this:

1. Existing N-S equations are great & we are only using artificial viscosity to overcome numeric round-off errors & due to scheme problems;

or,

2, We have missed something along the way & perhaps need to re-think a little.

---------

After my Control experiences, I find it very hard to believe that a complex non-linear system of equations such as N-S cannot contain at least one singularity.

diaw...

---------- Mani's comment: I have a basic question that is kind of answered in your discussion with Tom, but maybe you can comment to clarify.

Given that the stability is related to a grid Reynolds number, which is really a parameter that arises in the discretized problem and seems to have little to do with the continuous N-S equations, wouldn't you assume that any singularities you might encounter are really singularities of the discretized equations (not singularities of the N-S equations)?

If you agree, this goes back to my original point. The issue here is really the stability of your numerical scheme which more or less adequately approximates the continuous equations. The necessity of stabilization methods such as artificial dissipation, upwinding, a.s.o. is really a penalty you have to pay for discretization, because the discretized equations behave quite differently from the continuous ones (taw's comment on the physical reason's for upwinding are quite appropriate). So it's really about discretization and not about singularities of the N-S equations. Maybe this is what you meant, and I just misunderstood you.

Tom's comment: ...A good example of a discrete problem behaving very differently to the continuous one is to discretize the logistic equation using a forward Euler scheme. The resulting discrete equation possesses a period doubling cascade and chaos as the free parameter (timestep) is varied while the continous problem has no such behaviour. In the limit of the timestep going to zero the discrete problem approaches the continuous soultion but outside this range there is no reason for them to share the same behaviour.

diaw's reply: I have checked the simple Logistic equation (1st order in time)- 'Elementary Differential Equations & Boundary Value Problems' page 79-83, Boyce/DiPrima - At first glance, I find no apparent singularity. If anything in my system the point-of-interest would lie at the origin & thus, at this stage, would not concern me at all. I would imagine that weaving the non-linearities into the discrete scheme may also account for some of the problems folks find. I would imagine a non-linear loop would be imperative. In the simple form, these non-linear terms would look somewhat like a 'non-linear source term'.

Did you have a more complex version in mind than the simple version?

diaw...

>1. Existing N-S equations are great & we are only using artificial viscosity to overcome numeric round-off errors & due to scheme problems;

Let's leave the possibility of singularities aside for a second and see what we can do just by looking at the scheme.

I am still not sure you are any further than the good old Van-Neumann analysis. I think, if you do get further, your method should still agree with the basic findings of the linear analysis. Round-off errors are no problem, as long as the numerical scheme is stable. They are not the reason why we use artificial dissipation, and they are not the reason for instability (just the trigger). If you compare the discretized equations with the continuous equations, you can find a modified continuous equation, that is exactly described by your discrete model. Of course, this modified equation is not equivalent to the original PDE, but it should at least be consistent (i.e. if you increase the resolution, discretization errors have to go to zero). Studying this modified equation, you will see terms of diffusion and dispersion which do not appear in the original equations. These are the errors that show up because of discretization, and they very much depend on the scheme you choose (regardless of singularities of the N-S equations). The diffusion error is usually the bad guy, causing instability. Depending on the sign of the erroneous diffusion (positive or negative) your scheme will be either stable or unstable. The instability is simply caused by non-physical negative diffusion which lets any perturbation blow up. The diffusion error usually depends on the CFL number, and maybe other parameters (grid Reynolds number?). Artificial dissipation is added to counter the negative dissipation of your base scheme. Upwinding schemes are usually (positively) diffusive, sometimes excessively so. This may be a simplified view based on hyperbolic problems simpler than the N-S equations, and I am sure there may be additional caveats with N-S, but I still see the diffusion error as a main cause of problems, and that's simply due to discretization.

Now, I am sure you know all the above, and you can analyze your scheme in much the same way, but I still don't see how this fits into your view of instability due to singularities. If your scheme blows up, even though it is supposedly stable (according to the modified equation), you'll have a lot of other things to rule out, as I mentioned earlier, before you can conclude that the cause is in the continuous N-S equations.

Well put.

Hi Mani,

Thank you for your excellent, concise review... I add my thoughts...

------------------------

Mani: I am still not sure you are any further than the good old Van-Neumann analysis.

diaw: Von-Neumann analysis is a neat trick. Seems similar to 'Fourier mode' & 'einsantz' approaches. Is their perhaps a similarity to the Laplace Transform?

Actually what is the Laplace Transform? In my mind, at least, it appears to be no more than a 'weighting function'.

int(exp(-s*t)*f(t),t,0,t) => weights the original function over the whole domain, to be 'squeezed' to the origin, by applying increased damping as t grows larger. Very much like a FEM element interpolation function acting on the upstream node...

exp(+....) would perhaps act *away* from the origin?

If all singularities are 'squeezed' to zero by the 'differential transform' 'in the limit as dx tends to zero', then I imagine that exp(+...) would create a small perturbation a very small distance from the 'vanishing point'. The exp(+i...) allows the family of sinusoidal functions (-> Fourier series => approximate any function) to be included. We have now arrived at a compact form of Boyce/DiPrima's approach to explaining stability of the wave equation - p623-632. The study of the growth, or decay of this disturbance allows us to investigate the stability of the governing pde subject to its coefficients.

[Sidebar - how does the size of this small disturbance compare to the size of a 'computing element'?]

-----------------------

Mani: If you compare the discretized equations with the continuous equations, you can find a modified continuous equation, that is exactly described by your discrete model. Of course, this modified equation is not equivalent to the original PDE, but it should at least be consistent (i.e. if you increase the resolution, discretization errors have to go to zero).

diaw: But, surely we have then set about to solve the wrong governing equation?

This is my contention with the addition of artificial viscosity (upwinding etc) - leading to 'false diffusion' in the resuting flow field. In most of my early work in FVM, I used a blended schems of 95% central-discretisation: 5% upwinfing so that I could minimise the solution-smearing.

In the limit as 'dx->0', a correct numerical scheme must surely arrive back at the original pde?

Adding terms to the original equation is not equivalent to transform-solve-inv_transform process, which is where I hope to eventually end up.

----------------------

Mani: Studying this modified equation, you will see terms of diffusion and dispersion which do not appear in the original equations. These are the errors that show up because of discretization, and they very much depend on the scheme you choose (regardless of singularities of the N-S equations). The diffusion error is usually the bad guy, causing instability. Depending on the sign of the erroneous diffusion (positive or negative) your scheme will be either stable or unstable. The instability is simply caused by non-physical negative diffusion which lets any perturbation blow up. The diffusion error usually depends on the CFL number, and maybe other parameters (grid Reynolds number?). Artificial dissipation is added to counter the negative dissipation of your base scheme. Upwinding schemes are usually (positively) diffusive, sometimes excessively so. This may be a simplified view based on hyperbolic problems simpler than the N-S equations, and I am sure there may be additional caveats with N-S, but I still see the diffusion error as a main cause of problems, and that's simply due to discretization.

diaw: I concur that: positive diffusion error assists, while negative diffusion error makes trouble. I observe that the viscosity is reflected in the 'singularity' & hence any adjustment to system viscoisty will affect this beast. (Sorry to mention the word 'singularity' again The manipulation of viscosity is what upwinding seems to do.

----------------------

Mani: Now, I am sure you know all the above, and you can analyze your scheme in much the same way, but I still don't see how this fits into your view of instability due to singularities. If your scheme blows up, even though it is supposedly stable (according to the modified equation), you'll have a lot of other things to rule out, as I mentioned earlier, before you can conclude that the cause is in the continuous N-S equations.

diaw: (Refering to the simple example I mention earlier in this thread). In the example I was testing whether schemes do blow up at the 'singular point', or at some multiple of it. I have found that some schemes clearly blow up at a multiple (leading me to consider inbuilt numerical 'viscosity' in the scheme) - some did not blow up at all... Of interest is that in testing one such 'stable scheme', for flow over a cylinder over the critical Re, I found that the steady solver would still predict a steady flow-field at way beyond the point of transition to turbulence, whereas the scheme that did blow up in my test case began to oscillate more & more the closer I moved to this point. This scheme then also refused to provide a steady solution beyond the point of transition. The oscillation effect up to the point of transition can also be observed using the FeatFlow solvers - 'cc2d' in specific - with no upwinding.

My transformation is applied directly to the N-S equations themselves, in order to expose the underlying structure - leading to 'singularities'. I have then tested standard solvers against the supposed 'singularity/ies' in order to see if some respond.

I am currently using the insights I have gained to develop a new scheme with (hopefully) no spurious terms...

--------------- It has been very interesting work thus far & I trust that some good can come from all of this...

diaw...

correction,

int(exp(-s*t)*f(t),t,0,t) should read

int(exp(-s*t)*f(t),t,0,inf) for definition of Laplace Transform.

diaw...