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Jacobian Free-Newton Krylov Discontinuous Galerkin Method and Physics-Based Preconditioning for Nuclear Reactor Simulations

Description: Multidimensional, higher-order (2nd and higher) numerical methods have come to the forefront in recent years due to significant advances of computer technology and numerical algorithms, and have shown great potential as viable design tools for realistic applications. To achieve this goal, implicit high-order accurate coupling of the multiphysics simulations is a critical component. One of the issues that arise from multiphysics simulation is the necessity to resolve multiple time scales. For example, the dynamical time scales of neutron kinetics, fluid dynamics and heat conduction significantly differ (typically >1010 magnitude), with the dominant (fastest) physical mode also changing during the course of transient [Pope and Mousseau, 2007]. This leads to the severe time step restriction for stability in traditional multiphysics (i.e. operator split, semi-implicit discretization) simulations. The lower order methods suffer from an undesirable numerical dissipation. Thus implicit, higher order accurate scheme is necessary to perform seamlessly-coupled multiphysics simulations that can be used to analyze the “what-if” regulatory accident scenarios, or to design and optimize engineering systems.
Date: September 1, 2008
Creator: Park, HyeongKae; Nourgaliev, R.; Martineau, Richard C. & Knoll, Dana A.
Partner: UNT Libraries Government Documents Department

Jacobian-free Newton Krylov discontinuous Galerkin method and physics-based preconditioning for nuclear reactor simulations

Description: We present high-order accurate spatiotemporal discretization of all-speed flow solvers using Jacobian-free Newton Krylov framework. One of the key developments in this work is the physics-based preconditioner for the all-speed flow, which makes use of traditional semi-implicit schemes. The physics-based preconditioner is developed in the primitive variable form, which allows a straightforward separation of physical phenomena. Numerical examples demonstrate that the developed preconditioner effectively reduces the number of the Krylov iterations, and the efficiency is independent of the Mach number and mesh sizes under a fixed CFL condition.
Date: September 1, 2008
Creator: Park, HyeongKae; Nourgaliev, Robert R.; Martineau, Richard C. & Knoll, Dana A.
Partner: UNT Libraries Government Documents Department

Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

Description: In recent work (Nourgaliev, Liou, Theofanous, JCP in press) we demonstrated that numerical simulations of interfacial flows in the presence of strong shear must be cast in dynamically sharp terms (sharp interface treatment or SIM), and that moreover they must meet stringent resolution requirements (i.e., resolving the critical layer). The present work is an outgrowth of that work aiming to overcome consequent limitations on the temporal treatment, which become still more severe in the presence of phase change. The key is to avoid operator splitting between interface motion, fluid convection, viscous/heat diffusion and reactions; instead treating all these non-linear operators fully-coupled within a Newton iteration scheme. To this end, the SIM’s cut-cell meshing is combined with the high-orderaccurate implicit Runge-Kutta and the “recovery” Discontinuous Galerkin methods along with a Jacobian-free, Krylov subspace iteration algorithm and its physics-based preconditioning. In particular, the interfacial geometry (i.e., marker’s positions and volumes of cut cells) is a part of the Newton-Krylov solution vector, so that the interface dynamics and fluid motions are fully-(non-linearly)-coupled. We show that our method is: (a) robust (L-stable) and efficient, allowing to step over stability time steps at will while maintaining high-(up to the 5th)-order temporal accuracy; (b) fully conservative, even near multimaterial contacts, without any adverse consequences (pressure/velocity oscillations); and (c) highorder-accurate in spatial discretization (demonstrated here up to the 12th-order for smoothin-the-bulk-fluid flows), capturing interfacial jumps sharply, within one cell. Performance is illustrated with a variety of test problems, including low-Mach-number “manufactured” solutions, shock dynamics/tracking with slow dynamic time scales, and multi-fluid, highspeed shock-tube problems. We briefly discuss preconditioning, and we introduce two physics-based preconditioners – “Block-Diagonal” and “Internal energy-Pressure-Velocity Partially Decoupled”, demonstrating the ability to efficiently solve all-speed flows with strong effects from viscous dissipation and heat conduction.
Date: January 1, 2008
Creator: Nourgaliev, Robert; Theofanous, Theo; Park, HyeongKae; Mousseau, Vincent & Knoll, Dana
Partner: UNT Libraries Government Documents Department

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

Description: 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 singlepoint 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. The focus of this work is placed on fully-implicit, high-order spatiotemporal discretization based on the discontinuous Galerkin method solving the conservative form of the compressible Navier-Stokes equations. The method utilizes a local reconstruction procedure derived from weak formulation of the problem, which is inspired by the recovery diffusion flux algorithm of van Leer and Nomura [?] and by the piecewise parabolic reconstruction [?] in the finite volume method. The developed methodology is integrated into the Jacobianfree Newton-Krylov framework [?] to allow a fully-implicit solution of the problem.
Date: July 1, 2008
Creator: Park, HyeongKae; Nourgaliev, Robert; Mousseau, Vincent & Knoll, Dana
Partner: UNT Libraries Government Documents Department

Recovery Discontinuous Galerkin Jacobian-Free Newton-Krylov Method for All-Speed Flows

Description: A novel numerical algorithm (rDG-JFNK) for all-speed fluid flows with heat conduction and viscosity is introduced. The rDG-JFNK combines the Discontinuous Galerkin spatial discretization with the implicit Runge-Kutta time integration under the Jacobian-free Newton-Krylov framework. We solve fully-compressible Navier-Stokes equations without operator-splitting of hyperbolic, diffusion and reaction terms, which enables fully-coupled high-order temporal discretization. The stability constraint is removed due to the L-stable Explicit, Singly Diagonal Implicit Runge-Kutta (ESDIRK) scheme. The governing equations are solved in the conservative form, which allows one to accurately compute shock dynamics, as well as low-speed flows. For spatial discretization, we develop a “recovery” family of DG, exhibiting nearly-spectral accuracy. To precondition the Krylov-based linear solver (GMRES), we developed an “Operator-Split”-(OS) Physics Based Preconditioner (PBP), in which we transform/simplify the fully-coupled system to a sequence of segregated scalar problems, each can be solved efficiently with Multigrid method. Each scalar problem is designed to target/cluster eigenvalues of the Jacobian matrix associated with a specific physics.
Date: July 1, 2008
Creator: Park, HyeongKae; Nourgaliev, Robert; Mousseau, Vincent & Knoll, Dana
Partner: UNT Libraries Government Documents Department