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Re: Optimization of a finite volume differencing scheme for
multispecies transport problem
Posted:
Aug 2, 2008 4:59 PM
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On 1 août, 22:09, boulou...@gmail.com wrote:
> On 1 août, 12:40, Olin Perry Norton <mylastn...@icet.msstate.invalid>
> wrote:
>
>
>
> > boulou...@gmail.com wrote:
> > > I am working on a 3d finite volume scheme for an advection-diffusion-
> > > reaction problem involving a large number of chemical species (more
> > > than 60) and a large domain (an big lake for example). Since this
> > > scheme will be used on large problem, I want it to be as efficient as
> > > possible. The linear operators are splitted in 2 :
>
> > > (1) advection-diffusion is solved using a fully implicit finite volume
> > > discretisation with a multigrid method for solving the linear system
> > > of equations
> > > (2) chemistry is solved using a Runge-Kutta-Rosenbrock solver for
> > > stiff ODE.
>
> > > The transport (1) actually have the following form
>
> > > foreach specie in speciesList {
> > > construct_matrix();
> > > solve_linear_system();
> > > }
>
> > > and takes a lot of time on the computer.
>
> > > Assuming that diffusion coefficients are the same for all species, the
> > > whole fluid (including all species) should follow the same path during
> > > the transport. I wonder if it really necessairy to loop over all
> > > species and compute the transport several time. It is possible to
> > > compute the transport of the fluid once, and after reuse this
> > > calculation to the different species ?
>
> > > I would really appreciate suggestion or reference on this.
>
> > What you suggest is indeed possible.
>
> > Suppose that C(n) is the vector of concentrations of a certain
> > species at timestep n. This vector will have as many dimensions
> > as there are cells in your grid, and there will be a similar
> > vector for each species. I have used just one index, "n", to
> > indicate the timestep, but we could clearly add others to
> > indicate species and to identify grid location.
>
> > The advection-diffusion equation is linear, so, unless you
> > are doing something like flux-limiting, this concentration
> > vector at the next timestep, n+1, will be a linear function
> > of C(n), i.e.,
>
> > C(n+1) = T * C(n)
>
> > where T is a matrix. It is a square matrix -- the number of
> > entries in this matrix is the square of the number of grid points
> > you have. The matrix T takes a concentration profile
> > and diffuses and convects it by one timestep.
>
> > Clearly, since the advection and diffusion process is the same
> > for all species, once you have computed T for one species,
> > you can use the same T for all species.
>
> > Also note that, unless the diffusion coefficient or the flow
> > velocity in your lake change with time, you can continue to use
> > the same T for every timestep.
>
> > Mathematically, this is based on the fact that the advection-diffusion
> > equation is linear in the concentration variable -- if you're familiar
> > with Green's functions, this is basically a discretized version of
> > a Green's function.
>
> > The drawback I see is that ithis method would require solving for
> > and storing a large matrix -- if M is the number of gridpoints
> > in your problem, then T would be an MxM matrix. Perhaps there is
> > a clever way around this, or maybe it is manageable. Most of the
> > entries in T should be very close to zero.
>
> > Olin Perry Norton
>
> Thank you very much for these informations, really appreciated.
>
> If I understand, what you suggest is to do the following :
>
> 1) Suppose I use a fully implicit (backward Euler for example) or semi-
> implicit (Crank-nicholson) time integrating scheme. I will use a
> finite volume discretisation to create a linear system in the form
>
> A * c(n+1) = c(n)
>
> 2) Compute
>
> T = inverse(A)
>
> Using a finite volume discretisation, with an hybrid differencing
> scheme (central/upwind depending on the Peclet number) for the
> advective part, the matrix A will be diagonnally dominant and all
> entries are positive. Hence, the inverse of A exists and could be
> computer with an appropriate method (do you have any suggestions on
> this ?)
>
> 3) loop over all species and solve
>
> C(n+1) = T * C(n)
>
> for each one.
>
> If my understanding of your suggestion is correct, this would be a
> very elegant solution to my problem !
>
> If you have reference on this method or it's application in CFD, I
> would be interested to read them.
>
> Again thank you very much.
The problem with this approach, as you mentionned, is that if A is
sparse then A^-1 will almost surely be dense. Since I will need to
solve this system more than 60
times on a grid of 500x500x50 the inverse matrix will probably takes a
lot of memory.
Maybe saving the A=LU factorization for further use could be a better
idea. It is certainly less computationally expensive, but I am not
sure if L and U will be sparse.
If both matrices are sparce, I could save them and once I have solved
(LU) * c(n+1) = c(n)
for one specie, I could use them for another specie.
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