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What books for numerical solutions to PDEs

  1. Jun 16, 2015 #1

    DEvens

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    What are some good books (or other resources) on numerical methods of solving PDEs in 3 space and 1 time variable?

    I am interested both in finite element and finite volume methods. I could be interested in other methods but I don't know about them. I am interested in being able to take advantage of parallel computing. I have budget to buy some books. I have done quite a bit of 1-space 1-time numerical work.

    The specific equation I am most interested in is the diffusion equation with quite a few complications. It's transport of neutrons in a nuclear reactor. For example, there are many species diffusing (each is an energy group, as many as 39 in the full model), and these can change from one to the other. The system is changing over time as well as the concentration of species. There are many interesting time scales in the problem from potentially microseconds (if things get badly wrong) up to weeks to account for consumption of fuel. The boundary conditions are interesting.

    One book already on my to-buy list is Hamming "Numerical Methods for Scientists and Engineers."

    Any other suggestions?
     
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  3. Jun 17, 2015 #2

    DEvens

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    Sorry, I meant finite difference and finite volume methods.
     
  4. Jun 17, 2015 #3

    MathematicalPhysicist

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  5. Jun 18, 2015 #4

    Astronuc

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    Is one interested in thermal, epi-thermal or fast reactors, or all spectra? Whatever the spectrum, ideally the methods allow for steady-state operation with depletion and accumulation of fission products and transuranics, as well as transient conditions. In an LWR, the time scale of reactivity-insertion event is on the order of ms. One can review experiments from the French CABRI and Japanese NSRR RIA experiments. For steady-state, time steps are on the order of days, and depletion calculations are typically done for steady-state full power conditions, and generally not for power maneuvers.

    MIT's OpenMOC Method of Characteristics Code should be of interest - https://mit-crpg.github.io/OpenMOC/

    This is a reasonably good characterization of the problem.- https://en.wikipedia.org/wiki/Neutron_transport#Discretization_in_Deterministic_Methods

    See also theory of numerical methods for hyperbolic PDEs - http://www.aei.mpg.de/~rezzolla/lnotes/Hyperbolic_Pdes/hyperbolic_pdes_lnotes.pdf
     
  6. Jun 18, 2015 #5

    DEvens

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    The final target of this work will be, most of the time, power reactors for nuclear power stations. Or reactors in that general category. I am in Ontario, so most of the time I will want to apply this to CANDU reactors. Possibly a research reactor will sneak into the calculations, but it will tend to be a reactor used to test out fuel etc. for use in power reactors. So predominantly thermal.

    Mostly this is me trying to persuade the powers-that-be in my area that we should move into the era of modern computing. And that means I need to be up to date on what is possible and practical and useful.
     
  7. Jun 21, 2015 #6
    A nice introductory book on the FVM is Versteeg & Malalasekera's "An Introduction to Computational Fluid Dynamics - The Finite Volume Method". I have this book and I like it a great deal. Yet it may be a bit off from your niche of application.

    Coincidentally, I also have Hamming's "Numerical Methods for Scientists and Engineers". Frankly, I doubt it will be useful to you given the fields you mentioned as the ones of your interest (FDM, FVM, PDEs).
     
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