I am presently researching the role of eigenvalues, eigenvectors & their role in PDE's. I have read through a number of general Math texts, but am looking for something more in-depth with application to fluid PDE's.

I have performed some extremely interesting simulations & would like to check the background theory more thoroughly.

I would really value a few good pointers to advanced texts & papers.

Many thanks, diaw...

Further to my previous note... a few questions...

What role would the eigenvalues & eigen-vectors play in understanding the pde structure?

Would they have any physical meaning?

Are eigen-values/vectors able to be combined in any meaningful way?

diaw...

Computational Gas dynamics - Laney

gives a detailed physical explanation of the role of eigen values and vectors in the context of Euler's equation.Also most books have discussions that revolve around hyperbolic systems.

-H

Thanks very much, Harish for the useful reference & thoughts...

diaw...

An interesting result in a numeric experiment:

1) Eigen-values & eigen-vectors obtained for a system matrix A - with zero offset.

2) Under certain conditions, only real eigenvalues occur.

3) With a slight change in value of a system boundary condition & suddenly all eigenvaleus are in complex pairs - with no real eigenvalues.

This is intriguing.

diaw...

The rule of thumb in deciding the kind of pde from eigen values is

Elliptic: complex Hyperbolic:real and distinct

For Euler equation if you do a symbolic analysis of the system of equations for an isentropic flow you will get three eigen values corresponding to the left and right going acoustic wave travelling at speeds U+c and U-c and the vorticity wave travelling with the flow speed.But for a N-S system you would get a system of ellipctic-parabolic etc.. depending on the terms you include in your equation.

For a hyperbolic system eigen values can be viewed as the speed at which the information travels in the direction given by the eigen vectors.A similar explanation can be given for ellipctic systems.

-H

Thanks again Harish - exactly what I was looking for...

Now, if you had a system with hyperbolic governing equation - eg. flow around an object - would you expect to see more than 3 eigenvalues? In other words, if there is more than one wave present?

If the governing equation were such that only complex eigenvalues (complex pairs) are obtained - what physical significance can we attach to these?

diaw...

Diaw You might be interested in having a look at the book written by Harvard Lomax (Fundamentals of Computational Fluid Dynamics - a springer publication) Constantly uses the EigenVectors and Values to explain the CFD basics Good luck and Enjoy the reading.

An eigen value for each equation and an eigen vector indicating the direction.For an Euler system (assume 2-d) including energy equation you have 4 equations continuity,2 velocity and energy so there are four waves ie vorticity,entropy,acoustic left and right propagating.

%%If the governing equation were such that only complex eigenvalues (complex pairs) are obtained - what physical significance can we attach to these?

The system is elliptic.This can sometimes happen for a hyperbolic system too with non variable coefficients.

Consider a system u,t+Au,x=0 when A is variable of (u,x) then A may have complex roots at some points and real values at some points.

An interpretation can be given in terms of eigen values for navier stokes why some numerical algorithms cannot be used for both kinds of flows incompressible and compressible.

In the field of aeroacoustics the numerical methods revolve around being able to resolve the waves correctly and also not alter their dispersion relation.

There is a classical book by richtmeyer and morton which discusses this topic in detail.Also all these analysis are restricted to linear systems and extended heurestically to non linear systems.

Shock phenomenom can also be explained by looking at caracteristics.

-H

Ahmed wrote:

You might be interested in having a look at the book written by Harvard Lomax (Fundamentals of Computational Fluid Dynamics - a springer publication) Constantly uses the EigenVectors and Values to explain the CFD basics Good luck and Enjoy the reading.

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diaw's reply:

Greetings Ahmed - I hope you are well.

Thanks so much for that reference... I'll get hold of it. This research phase is becoming extremely interesting.

Harish wrote:

An eigen value for each equation and an eigen vector indicating the direction.For an Euler system (assume 2-d) including energy equation you have 4 equations continuity,2 velocity and energy so there are four waves ie vorticity,entropy,acoustic left and right propagating.

diaw's reply:

This is fine for 2-d equations positioned in open space, but, what effects do boundaries impose on the number of available/possible waves in the system. Think of a box with flow entering from the left, exiting from the right, with fixed boundaries top & bottom - ie. a duct.

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diaw: %%If the governing equation were such that only complex eigenvalues (complex pairs) are obtained - what physical significance can we attach to these?

Harish:

The system is elliptic.This can sometimes happen for a hyperbolic system too with non variable coefficients.

Consider a system u,t+Au,x=0 when A is variable of (u,x) then A may have complex roots at some points and real values at some points.

diaw:

From your reasoning for hyperbolic governing-equations, I would infer that parabolic systems could also show similar behaviour. Interesting.

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Harish:

An interpretation can be given in terms of eigen values for navier stokes why some numerical algorithms cannot be used for both kinds of flows incompressible and compressible.

diaw:

This is a very useful observation - thanks.

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Harish:

In the field of aeroacoustics the numerical methods revolve around being able to resolve the waves correctly and also not alter their dispersion relation.

There is a classical book by richtmeyer and morton which discusses this topic in detail.Also all these analysis are restricted to linear systems and extended heurestically to non linear systems.

diaw:

I would imagine, from the term aeroacoustics, that you are refering to high-speed, inviscid flows? Are they using Euler-type equations? I'll research the reference. Thanks very much.

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Harish:

Shock phenomenom can also be explained by looking at caracteristics.

diaw:

Now we are indeed getting interesting. How would the values of the eigenvalues be used to infer information about the relevant system characteristics?

Thanks so much for your very kind discussion & references.

diaw...

%%%% This is fine for 2-d equations positioned in open space, but, what effects do boundaries impose on the number of available/possible waves in the system. Think of a box with flow entering from the left, exiting from the right, with fixed boundaries top & bottom - ie. a duct.

To study 2-d we need to look into details about what information is carried by the characteristics.Consider the Euler equations.A possible set of characteristics are u+p and u-p u being the x-velocity and p being the pressure.For waves reflected from the wall u=0 and hence for all the points which lie along the characteristics originating from the wall you will see u=0 and calculate the value from u+p=const along that characteristic.A detailed explanation can be found in chapter 6 of tanehill,pletcher and anderson.

The characteristics are not so important for parabolic systems because the speed of travel is infinite and hence the travel of information along characteristics is not clearly visible.Consider a diffusion equation with a sinusoidal IC.It would be impossible to do an inverse problem from say t=9 s to t=0 because the original information is lost.

Aeroacoustics is a generic term for compressible flows where the question asked is what is the sound emitted in the far field due to the effects of unsteady velocity field.They usually use Euler type equations because the effect of viscosity is almost negligible.

%Now we are indeed getting interesting. How would the values of the eigenvalues be used to infer information about the relevant system characteristics?

Chapter two of tanehill anderson and pletcher discusses this question in detail for the three systems.

Thanks Harish for that excellent reply. Ive got my afternoon's reading material cut out for me... Thanks for the detailed chapter references.

I'll reply back in more detail after I've read the sections in Tannehill et al.

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A point of question on parabolic equations.

Harish wrote:

The characteristics are not so important for parabolic systems because the speed of travel is infinite and hence the travel of information along characteristics is not clearly visible.

diaw:

The N-S are often classified as parabolic in nature. Would this imply that infinite speed of travel occurs? Is this the unique solution, or a particular solution.

My research observations on N-S from both a mathematical solution & simulation viewpoint is one of decaying/growing characterisitics, with a range of finite wave-speeds possible.

Thanks again for your exceptional contribution.

diaw... (Des Aubery)

%%The N-S are often classified as parabolic in nature. Would this imply that infinite speed of travel occurs? Is this the unique solution, or a particular solution.

The N-S is referred as parabolic when the inertia term is negligible like in BL and hence it becomes more like a system of diffusion equations and hence you can say that wave speed of infinity.

%% My research observations on N-S from both a mathematical solution & simulation viewpoint is one of decaying/growing characterisitics, with a range of finite wave-speeds possible.

When the effect of viscosity and thermal conductivity is taken into account the waves can become decaying/growing.But in most cases their effect on the information travel can be neglected as long as the wavelength is much larger than mean free path.The easiest way to look at the wave nature of a system is doing the following.For eg. for Euler's equations write the system in the form

u,t+A u,x + B u,y + C =0 and find the eigen values & vectors of A or B symbolically using matlab,maxima or mathematica, it can give a great deal of information about the acoustics locked into the equation.

-H

diaw wrote:

The N-S are often classified as parabolic in nature. Would this imply that infinite speed of travel occurs? Is this the unique solution, or a particular solution.

Harish:

The N-S is referred as parabolic when the inertia term is negligible like in BL and hence it becomes more like a system of diffusion equations and hence you can say that wave speed of infinity.

diaw:

Thanks. So, parabolic would imply diffusion-dominated, rather than convection-dominated. Good.

I've seen the 'wave speed of infinity' logic related to the diffusion equation, in a previous thread & wonder how it was derived. To my mind, it seems to be somewhat of a mathematical 'fiction', since, in practice no communication can travel faster than light.

A standing-wave solution can be considered for the diffusion (heat) equation ie. zero wave speed, with time growth/decay of the standing wave. Think of a bar with an initial temperature distribution which decays/grows with time, depending on the source term/forcing function.

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diaw:

My research observations on N-S from both a mathematical solution & simulation viewpoint is one of decaying/growing characterisitics, with a range of finite wave-speeds possible.

Harish:

When the effect of viscosity and thermal conductivity is taken into account the waves can become decaying/growing.But in most cases their effect on the information travel can be neglected as long as the wavelength is much larger than mean free path.The easiest way to look at the wave nature of a system is doing the following.For eg. for Euler's equations write the system in the form

u,t+A u,x + B u,y + C =0 and find the eigen values & vectors of A or B symbolically using matlab,maxima or mathematica, it can give a great deal of information about the acoustics locked into the equation.

diaw:

Excellent point. Now, could I press you a little further on the last line where you refer to 'the eigen values & vectors of A or B ... can give a great deal of information about the acoustics locked into the equation'.

This is exactly where I am heading. Can you perhaps elaborate a little further on that point - perhaps an example?

Thanks again, Harish, for your very informative reply.

diaw:

Thanks. So, parabolic would imply diffusion-dominated, rather than convection-dominated. Good.I've seen the 'wave speed of infinity' logic related to the diffusion equation, in a previous thread & wonder how it was derived. To my mind, it seems to be somewhat of a mathematical 'fiction', since, in practice no communication can travel faster than light. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55 From the continuum point of view the speed of sound is infinity.For parabolic system it is assumed information is transmitted instantaneously to every other point in the domain.

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diaw:

Excellent point. Now, could I press you a little further on the last line where you refer to 'the eigen values & vectors of A or B ... can give a great deal of information about the acoustics locked into the equation'.

This is exactly where I am heading. Can you perhaps elaborate a little further on that point - perhaps an example? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% Consider the linearised Euler equation used in acoustics to derive the waveequation( can be found in any acoustics books ).If you write it in a system form it can be shown that the eigen values are u+c u-c u for x direction and similarly for y direction.Since they are real and distinct the system is hyperbolic which is the case with wave equation.

Unfortunately this analysis cannot be extended to the full N-S.The non linearity of the system prevents makes it hard.But there are some arguments for a system of equation with no convection term.It is said that such a system can become parabolic for real and distinct eigen values.Tanehill refers it to but does not go into the details of the method.The million dollar N-S baby of courant institute is still up for grab .

-Harish

"Unfortunately this analysis cannot be extended to the full N-S.The non linearity of the system prevents makes it hard.But there are some arguments for a system of equation with no convection term.It is said that such a system can become parabolic for real and distinct eigen values.Tanehill refers it to but does not go into the details of the method.The million dollar N-S baby of courant institute is still up for grab "

Actually the unsteady NS equations are parabolic - this has been known for some considerable time now. The convection terms play no real role in the clasification due to the fact they are lower order than the diffusion terms; i.e. d^2/dx^2 dominates d/dx - this is why the system is stiff when the coefficient of the diffusion term is small (it's also why, strictly speaking, the NS equations are quasi-linear rather than nonlinear).

The million dollar question is about the regularity, for all time, of the solution to the NS equations.

diaw - you could try the new book

"Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators" by L.N. Trefethen and M. Embree.

Greetings Tom,

Thanks so much for your reply - it is always very much appreciated.

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Harish:

"Unfortunately this analysis cannot be extended to the full N-S.The non linearity of the system prevents makes it hard.But there are some arguments for a system of equation with no convection term.It is said that such a system can become parabolic for real and distinct eigen values.Tanehill refers it to but does not go into the details of the method.The million dollar N-S baby of courant institute is still up for grab "

Tom:

Actually the unsteady NS equations are parabolic - this has been known for some considerable time now. The convection terms play no real role in the clasification due to the fact they are lower order than the diffusion terms; i.e. d^2/dx^2 dominates d/dx - this is why the system is stiff when the coefficient of the diffusion term is small (it's also why, strictly speaking, the NS equations are quasi-linear rather than nonlinear).

diaw:

Would this always be the case? As an example, take a fluid like water with a kinematic viscosity some 18 times smaller than that of air. At this point, the 'coefficients' for the second-order spatial terms become extremely small & their influence must surely tend towards small - leaving a proportionally larger convection-term influence. The N-S now begin to approach the Euler Equations assymptotically.

Where kinematic viscosity is large, the convection term would certainly dominated by the diffusion terms & the N-S would seem to approach a ?parabolic? situation. Where kinematic viscosity is tiny, as in real fluids, is where the game begins to get tricky & the equations move closer to a ?elliptic? (M<<1) form. (Please correct my terminology if I've incorrectly stated the forms).

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Tom:

diaw - you could try the new book

"Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators" by L.N. Trefethen and M. Embree.

diaw:

Thanks so much for that reference - I'll get it ordered.

diaw...

diaw: Would this always be the case? As an example, take a fluid like water with a kinematic viscosity some 18 times smaller than that of air. At this point, the 'coefficients' for the second-order spatial terms become extremely small & their influence must surely tend towards small - leaving a proportionally larger convection-term influence. The N-S now begin to approach the Euler Equations assymptotically.

This is the old problem - let nu_k for k =1,2,... denote values of the viscosity with nu_k -> 0 as k -> infinity. In the NS aquations does the sequaence of solutions u_k converge to the solution of the Euler equations?

The answer to this is generally no! Think about the Kutta condition for flow past an aerofoil and ask yourself where the "miraculous" value of the circulation comes from.

In the case of classification of the equations "size does not matter" it is simply the fact that the terms are there in the first place (another way to think of this is that the Navier-Stokes equations need more boundary conditions than the Euler equations - does the effect of these extra bcs go away as the viscosity is reduced? where have the extra bcs gone?).

Try solving Burgers equation for a range of decreasing values of the viscosity. You should find that the decay of the shock does not depend upon the viscosity in the limit nu->0. However the viscosity is controlling the decay of the shock!

diaw: Where kinematic viscosity is large, the convection term would certainly dominated by the diffusion terms & the N-S would seem to approach a ?parabolic? situation. Where kinematic viscosity is tiny, as in real fluids, is where the game begins to get tricky & the equations move closer to a ?elliptic? (M<<1) form. (Please correct my terminology if I've incorrectly stated the forms).

You really need to put this into the context of a Reynolds number: from the observation that the viscosity is small it does not follow that the inertia terms are bigger (think of very slow flow or a very thin pipe).

As I've already stated its not the size of these terms that is important - it is there implications upon what is a well defined problem. In general the Euler equations are hyperbolic (if I recall correctly it's the steady state where the distinction between elliptic (M<1) and hyperbolic (M>1) arises).

You may want to check the book out in a library to check its relevance to your problem before buying it.

Thanks Tom for your excellent comments.

Tom wrote:

In the case of classification of the equations "size does not matter" it is simply the fact that the terms are there in the first place (another way to think of this is that the Navier-Stokes equations need more boundary conditions than the Euler equations - does the effect of these extra bcs go away as the viscosity is reduced? where have the extra bcs gone?).

Try solving Burgers equation for a range of decreasing values of the viscosity. You should find that the decay of the shock does not depend upon the viscosity in the limit nu->0. However the viscosity is controlling the decay of the shock!

diaw: (*new*)

I know that intuitively you are correct in your observations. I had been grappling with these concepts & consequently I have developed a very simple graphical way to map pde's. Each pde has a 'shape' defined by the equation terms & this allows a geometric model to be derived. If one then takes the viscosity (or velocity) to the limit, the 'shape' swells/shrinks/moves - but basically still retains its fundamental 'shape'.

So, even though the viscosity would tend to limit zero, the term does not seems to 'jump' out of the equation - the links still remain. You are perfectly correct. Thanks for that prompt.

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diaw: Where kinematic viscosity is large, the convection term would certainly dominated by the diffusion terms & the N-S would seem to approach a ?parabolic? situation. Where kinematic viscosity is tiny, as in real fluids, is where the game begins to get tricky & the equations move closer to a ?elliptic? (M<<1) form. (Please correct my terminology if I've incorrectly stated the forms).

Tom:

You really need to put this into the context of a Reynolds number: from the observation that the viscosity is small it does not follow that the inertia terms are bigger (think of very slow flow or a very thin pipe).

As I've already stated its not the size of these terms that is important - it is there implications upon what is a well defined problem.

diaw:

My comment above would also be relevant here. You are correct in the fact that the terms are there in the first place is what is important.

*** A thought about numerical simulations... Is there not a danger that with each term being treated seperately, that the numeric size of some terms could cause them to simply drop out of the solution - in violation of what was mentioned above? In other words, perhaps the one-shoe-fits-all numeric algorithm may not be such a good idea after all? Perhaps the correct 'shape' algorithm needs to be also considered for the 'shape' of problem being solved?

My problem is that the area I am currently researching seems to be flipping me between two forms - parabolic & elliptic - at least as it appears at the moment. I'm really enjoying this...

Once again, thanks Tom for your excellent input. You always leave me with much food-for-thought.

diaw...