56,912 research outputs found
The Analytic Coarse-Mesh Finite Difference Method for Multigroup and Multidimensional Diffusion Calculations
In this work we develop and demonstrate the analytic coarse-mesh finite difference (ACMFD) method for multigroup - with any number of groups - and multidimensional diffusion calculations of eigenvalue and external source problems. The first step in this method is to reduce the coupled system of the G multigroup diffusion equations, inside any homogenized region (or node) of any size, to the G independent modal equations in the real or complex eigenspace of the G Ăâ G multigroup matrix. The mathematical and numerical analysis of this step is discussed for several reactor media and number of groups.
As a second step, we discuss the analytical solutions in the general (complex) modal eigenspace for one-dimensional plane geometry, deriving the generalized Chao's relation among the surface fluxes and the net currents, at a given interface, and the node-average fluxes, essential in the ACMFD method. We also introduce here the treatment of heterogeneous nodes, through modal interface flux discontinuity factors, and show the analytical and numerical application to core-reflector problems, for a single infinite reflector and for reflectors with two layers of different materials.
Then, we address the general multidimensional case, with rectangular X-Y-Z geometry considered, showing the equivalency of the methods of transverse integration and incomplete expansion of the multidimensional fluxes, in the real or complex modal eigenspace of the multigroup matrix. A nonlinear iteration scheme is implemented to solve the multigroup multidimensional nodal problem, which has shown a fast and robust convergence in proof-of-principle numerical applications to realistic pressurized water reactor cores, with heterogeneous fuel assemblies and reflectors
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
Cosmological Radiation Hydrodynamics with ENZO
We describe an extension of the cosmological hydrodynamics code ENZO to
include the self-consistent transport of ionizing radiation modeled in the
flux-limited diffusion approximation. A novel feature of our algorithm is a
coupled implicit solution of radiation transport, ionization kinetics, and gas
photoheating, making the timestepping for this portion of the calculation
resolution independent. The implicit system is coupled to the explicit
cosmological hydrodynamics through operator splitting and solved with scalable
multigrid methods. We summarize the numerical method, present a verification
test on cosmological Stromgren spheres, and then apply it to the problem of
cosmological hydrogen reionization.Comment: 14 pages, 3 figures, to appear in Recent Directions in Astrophysical
Quantitative Spectroscopy and Radiation Hydrodynamics, Ed. I. Hubeny,
American Institute of Physics (2009
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