Diffusion approximations to radiation transport feature a nonlinear conduction coefficient that leads to formation of a sharp front, or Marshak wave, under suitable initial and boundary conditions. The front can vary several orders of magnitude over a very short distance. Resolving the shape of the Marshak wave is essential, but using a global fine mesh can be prohibitively expensive. In such circumstances it is natural to consider using adaptive mesh refinement (AMR) to place a fine mesh only in the vicinity of the propagating front. In addition, to avoid any loss of accuracy due to linearization, implicit time integration should be used to solve the equilibrium radiation diffusion equation. Implicit time integration on AMR grids introduces a new challenge, as algorithmic complexity must be controlled to fully realize the performance benefits of AMR. A Newton-Krylov method together with a multigrid preconditioner addresses this latter issue on a uniform grid. A straightforward generalization is to use a multilevel preconditioner that is tuned to the structure of the AMR grid, such as the fast adaptive composite grid (FAC) method. We describe the resulting Newton-Krylov-FAC method and demonstrate its performance on simple equilibrium radiation diffusion problems.
Introduction.Radiation transport plays an important role in numerous fields of study, including astrophysics, laser fusion, combustion applications, atmospheric dynamics, and medical imaging. When photon mean free paths are much shorter than characteristic length scales, a diffusion approximation provides a reasonably accurate description of radiation penetrating from a hot source to a cold medium. This approximation features a nonlinear conduction coefficient that leads to formation of a sharp front, in which the solution can vary several orders of magnitude over a very short distance. The shape of the front can be very complex as it interacts with different materials having different conduction properties. Resolving these localized features with a global fine mesh can be prohibitively expensive. It is natural to consider reducing the cost of accurately resolving these fronts by using adaptive mesh refinement (AMR), which concentrates computational effort by increasing spatial resolution only locally.Classical solution techniques for equilibrium radiation diffusion use a linearized conduction coefficient to avoid the expense of solving a system of nonlinear equations at each time step. This introduces a first-order error in time that precludes effective use of higher-order time integration methods, and requires small time steps to maintain time accuracy. Analytic and computational results that demonstrate degradation in time accuracy associated with linearization in the presence of strong nonlinear coefficients can be found in [24,15]. Such effects can be avoided by using im-
This paper evaluates the performance of multiphysics coupling algorithms applied to a light water nuclear reactor core simulation. The simulation couples the k-eigenvalue form of the neutron transport equation with heat conduction and subchannel flow equations. We compare Picard iteration (block Gauss-Seidel) to Anderson acceleration and multiple variants of preconditioned Jacobian-free Newton-Krylov (JFNK). The performance of the methods are evaluated over a range of energy group structures and core power levels. A novel physics-based approximation to a Jacobian-vector product has been developed to mitigate the impact of expensive on-line cross section processing steps. Numerical simulations demonstrating the efficiency
The performance of object-oriented applications in scientific computing often sufTersfrom the inefficient use of high-level abstractions provided by underlying libraries. Since these library abstractions are not part of the programming language itself there is no compiler mechanism to respect their semantics and thus to perform appropriate optimizations, e.g., array semantics within object-oriented array class libraries which permit parallel optimizations inconceivable to the serial compiler. We have presented the ROSE inik%ructure as a tool for automatically generating library-specific preprocessors. These preprocessors can perform sematics-baaed source-to-source transformations of the application in order to introduce high-level code optimizations. In this paper we outline the design of ROSE and focus on the discussion of various approaches for specifying and processing complex source code transformations. These techniques are supposed to be as easy and intuitive as possible for the ROSE users, i.e. for the designers of the library-specific preprocessors.
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