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Dan October 4, 2008 05:16

OpenFoam & Sobolev spaces
 
Hi,

(i am not expert in theory of FVM) i want to know that which sobolev space is available in OpenFoam?

it seems that H^1 space should be commonly available as at least weak formulation of transport equations needs at least square integrability of function and its gradient

but if we need a C^1 space (i.e., global continuty of field derivative), is it available?

if we need local hessian of field variable, is it possible to compute in in FVM (in classic lagrange based FEM, it is not possible, of course a simple way is to use post-process to recover higher order quantity)?

in FEM computation these spaces are very clear, but i do not know any reference that address them in FVM, can someone suggest me a reference?


unimportant October 4, 2008 10:07

Re: OpenFoam & Sobolev spaces
 
FVM formulation is obtained through weak formulation by using the test functions that guarantee week partial derivatives of order 1 while preserving the square integrability. In other words, FVM formulation guarantees generalized derivatives that are equivalent to delta function and solutions that are equivalent to Heaviside function. The particular notion of generalized derivatives and associated spaces is really only used to demonstrate that hyperbolic equations that dominate compressible fluid dynamics lead to week solutions that observe basic conservation principles. Unlike FEM, test functions do not appear directly in the formulation of the discrete problem and their presence is not obvious. However, they are implicitly present in any FVM formulation as the whole formulation represents a week form of equations. Problems such as selection of the appropriate week solution is solved usually using the entropy arguments but that is not the topic of this discussion. In the case of incompressible flows without discontinuities, FVM formulation yields C1 continuity. However, it is worth knowing that in FVM second order discretization is always used for the smooth part of the solution thus satisfying C1 continuity requirements by using even smoother function than it is required. Of course, in the presence of the discontinuity, generalized derivatives are recovered by a special procedure usually practiced through so called "upwind formulation". To answer the question directly, Hessians with respect to spatial coordinates can be computed in the smooth part of the domain whereas Hessians with respect to flow variables for the purpose of iterative and optimization algorithms can (almost) always be computed.

FOAM implements FVM principles consistently and it allows you to compute generalized and classical solutions while computing gradients and Hessians is also possible. FOAM also has set of limiters that guarantee bounded solutions that are second order away from the discontinuities. I would suggest for mathematically inclined reader to check the theory of discontinuous Galerkin methods if more insight in theory is required but other than for formal proofs, this is not truly required. As in the case of many other methods, FVM has proven to be very reliable and valuable method in CFD and its properties are well understood but many many lack formal mathematical proof. This is not a show-stopper for the use of FVM in practical computations, however.

In summary, FVM solutions are usually of second order away from the discontinuity and they are of first order in the presence of discontinuity.


Dan October 4, 2008 12:57

Re: OpenFoam & Sobolev spaces
 
hi unimportant,

lot of thanks for your comments :)

> FVM formulation guarantees generalized derivatives that are equivalent to delta function

please clarify more, delta function implies to me the lowe regularity, like a Borel measure which is not like a proper function to be amenable for numerical computing

> generalized derivatives

what this mean? do you mean higher order derivatives as well?

> and solutions that are equivalent to Heaviside function

please give more details about this

you say about C^1 continuty of solution far from discontinuty, how we have this: is it due to continuty of flux accross cell faces?

then what is the related sobolev space our test function (consider weak formulation)? is it an H^2 function?

do you know some reference in this regard?

about DG, it seems that there is some similarity between DG and FVM, consider cell centered FVM (and linear flux interpolation, like OF), is there any DG which is equivalent with it?

probably we should consider lowest order, but does it ensure C^1 continuty?

about DG which book you recommned?

unimportant October 4, 2008 14:07

Re: OpenFoam & Sobolev spaces
 
>> FVM formulation guarantees generalized derivatives that >>are equivalent to delta function

>please clarify more, delta function implies to me the lowe >regularity, like a Borel measure which is not like a proper >function to be amenable for numerical computing

What I mean here is that if you take a derivative of the weak solution that represent a discontinuity, the result will be a delta function. Therefore, FVM solutions due to their weak formulation will seek functions only within the class of solutions whose derivatives behave this way. Whatever functional space is associated with them, they have to have these properties at the minimum.

>> generalized derivatives

>what this mean? do you mean higher order derivatives as >well?

Here I mean weak derivatives. Any good book on PDEs will have the definition of weak derivatives. One good book is "Partial Differential Equations" by Wloka (Cambridge Univ. Press)

>> and solutions that are equivalent to Heaviside function

>please give more details about this

Discontinuous solutions in FVM formulation are equivalent to Heaviside function. Shock waves, slip lines, etc. can be seen as Heaviside functions. Their derivatives are delta functions.

>you say about C^1 continuty of solution far from >discontinuty, how we have this: is it due to continuty of >flux accross cell faces?

Conservative (flux) variables are always continuous across the discontinuities. However, here I even think of primitive variables that can have C1 continuity because all of these functions are piecewise and the solver always knows where the discontinuity is. This is what I meant by a discretization methods that are capable of "upwinding". This is a bread and butter of CFD and FVM.

>then what is the related sobolev space our test function >(consider weak formulation)? is it an H^2 function?

>do you know some reference in this regard?

"Hyperbolic Systems of Conservation Laws and the mathematical Theory of Shock Waves" by Peter D. Lax. There you will find many answers to you questions.

>about DG, it seems that there is some similarity between DG >and FVM, consider cell centered FVM (and linear flux >interpolation, like OF), is there any DG which is >equivalent with it?

>probably we should consider lowest order, but does it >ensure C^1 continuty?

>about DG which book you recommned?

Bernardo Cockburn - search the web for references...

Dan October 4, 2008 15:50

Re: OpenFoam & Sobolev spaces
 
thanks for continuing

first forget about any discontinuty (shock, etc.) and assume a physically smooth solution)

but it would be better to talk in a more rigorous mathematical framework

in classic FEM we also knowe concept of weak solution. but let me to clear bit about weak solution, in this language, when we solve 2nd order pde, we actually need a C^2 space (or at least H^2) but we use weak formulation and so we can use more larger space, e.g., H^1 space, then our derivative is L^2, so it is not essentialy continuous without regard to field discontinuty you talk about, it is immediate result from our formulation. but this derivative is more regular than a delta function

i see some contradiction in your speech, maybe my miss understanding. from one side you say about C^1 solution of FVM and in the other side you say derivatives of solution are delta distribution property (delta distribution has the worst regularity)

i expect regularity of derivative more than L^2 from FVM

btw, thanks again for your contribution

other comments is wellcome!



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