Dana A. Knoll

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry

Abstract The use of the Jacobian-free Newton-Krylov (JFNK) method within the context of nonlinear diffusion acceleration (NDA) of source iteration is explored. The JFNK method is a synergistic combination of Newton’s method as the nonlinear solver and Krylov methods as the linear solver. JFNK methods do not form or store the Jacobian matrix, and Newton’s method is executed via probing the nonlinear discrete function to approximate the required matrix-vector products. Current application of NDA relies upon a fixed-point, or Picard, iteration to resolve the nonlinearity. We show that the JFNK method can be used to replace this Picard iteration with a Newton iteration. The Picard linearization is retained as a preconditioner. We show that the resulting JFNK-NDA capability provides benefit in some regimes. Furthermore, we study the effects of a two-grid approach, and the required intergrid transfers when the higher-order transport method is solved on a fine mesh compared to the low-order acceleration problem.

A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems

We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2,1,3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10,9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21,20]. The set of equations studied here constitute a base model for radiation hydrodynamics.

On physics-based preconditioning of the Navier-Stokes equations

We develop a fully implicit scheme for the Navier-Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton-Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M~10^-^4) with significant heat flux as well as intermediate speed (M~1) flows efficiently by following dynamical (advective) time scales of the problem.

Recovery Discontinuous Galerkin Jacobian-free Newton-Krylov Method for all-speed flows

There is an increasing interest to develop the next generation simulation tools for the advanced nuclear energy systems. These tools will utilize the state-of-art numerical algorithms and computer science technology in order to maximize the predictive capability, support advanced reactor designs, reduce uncertainty and increase safety margins. In analyzing nuclear energy systems. we are interested in compressible low-Mach number, high heat flux flows with a wide range of Re, Ra, and Pr numbers. Under these conditions, the focus is placed on turbulent heat transfer, in contrast to other industries whose main interest is in capturing turbulent mixing. Our objective is to develop single-point turbulence closure models for large-scale engineering CFD code, using direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) tools, requireing very accurate and efficient numerical algorithms.

Multiphysics Analysis of Spherical Fast Burst Reactors

Abstract Coupling neutronics to thermomechanics is important for the analysis of fast burst reactors because the criticality and safety study of fast burst reactors depends on the thermomechanical behavior of fuel materials. For instance, the shutdown mechanism or the transition between supercritical and subcritical states is driven by the fuel material expansion or contraction. The material expansion is due to the temperature gradient that results from fission power. In this paper, we introduce a numerical model for coupling of neutron diffusion and thermomechanics in fast burst reactors. The goal is to have a better understanding of the relation between the reactivity insertion and the thermomechanical response of fuel materials. We perform a nondimensional analysis of the coupled system that provides insight into the behavior of the transient. We also provide a semianalytical solution model to the coupled system for partial verification of our numerical solutions. We studied material behavior corresponding to different levels of reactivity insertion.