A Multigroup diffusion solver using pseudo transient continuation for a radiation-hydrodynamic code with patch-based AMR Page: 4 of 41
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(r = 15 cm) suspended in air at STP. For Y = 11 kT, we find that gray
radiation diffusion and MGD produce similar results. However, if Y = 1
MT, the two packages yield different results. Our large Y simulation con-
tradicts a long-standing theory and demonstrates the inadequacy of gray
This paper describes a numerical method to solve the radiation multigroup
diffusion (MGD) equations. Two themes are presented. One is the scheme
itself. We add Pseudo Transient Continuation (ftc) to the familiar "fully im-
plicit" method of Axelrod et al . The second theme is code-specific. Our
MGD solver is embedded in a multidimensional, massively parallel, Eulerian
radiation-hydrodynamic code, which has patch-based, time-and-space Adaptive
Mesh Refinement (AMR) capability. Our code's AMR framework stems from
the Berger and Oliger idea  developed for hyperbolic, compressible hydrody-
namic schemes. The idea was expanded by Almgren et al  and applied to
the type of elliptic solvers required for the incompressible equations of Navier-
Stokes. Howell and Greenough  applied the Almgren et al framework to the
scalar, parabolic "gray" radiation diffusion equation, thereby creating the start
of our radiation-hydrodynamic code.
The AMR framework works as follows. A domain, referred to as the "coarse"
or LO level, is discretized using a uniform, coarse spatial mesh size h,.1 After
advancing with a timestep Ate, the result is scanned for possible improvement.
One may refine subregions containing a chosen material, at material interface(s),
or at shocks, etc. Whatever refinement criteria are used, after the subdomains
are identified, specific routines define a collection of "patches," which cover the
subdomains. In two dimensions, the patches are unions of rectangles; in 3D,
they are unions of hexahedra. The patches need not be connected, but they
must be contained within the coarse level. The patches denote the "fine" or
L1 level and are discretized with a uniform, spatial mesh size hf. A typical
refinement ratio he/hf equals two, but higher multiples of two are also allowed.
Because the original framework was designed for temporally explicit hyper-
bolic schemes, At, is restricted by a CFL condition. This implies a similar
restriction for the L1 level timestep Atf. For the case, he/hf = 2, level L1
time-advances twice using At1 = At,/2. Boundary conditions for level L1 are
supplied as follows. Wherever level L1 extends to the physical boundary, the
level uses the conditions prescribed by the problem. Portions of level L1's bound-
ary which lie inside the physical domain have conditions prescribed by time and
space interpolated data obtained from the LO solution. For diffusion equations,
these conditions are of Dirichlet type. The numerical solution consists of both
coarse and fine grid results. Unfortunately, as it stands, the composite solution
does not guarantee conservative fluxes across the level boundaries. To maintain
1In multiple dimensions, coordinates have their own mesh spacing.
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Shestakov, A I & Offner, S R. A Multigroup diffusion solver using pseudo transient continuation for a radiation-hydrodynamic code with patch-based AMR, article, September 21, 2006; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc894060/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.