V.A. Mousseau

On balanced approximations for time integration of multiple time scale systems

The effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis (MEA). First, MEA is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation (ODE), and in a linear partial differential equation (PDE). Several time discretizations of the ODE and PDE are considered, and the resulting truncation terms are compared analytically and numerically. It is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing (fast) time scales is employed. Many of the issues uncovered on the simple problems are shown to persist in more realistic applications. Specifically, several differencing schemes using linearization and/or time splitting are applied to problems in nonequilibrium radiation-diffusion, magnetohydrodynamics, and shallow water flow, and their solutions are compared to those using balanced time integration methods.

An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows

In atmospheric flow situations typical of a small-scale atmospheric thermal, a separation of time scales exists between the fast sound wave time scale and the advective time scale. Atmospheric models have been designed to take advantage of this disparity of time scales with numerical approaches such as the semi-implicit or split-explicit approach being used to efficiently step over the fast sound waves. Some of these numerical approaches are first order in time. To improve accuracy over these methods, a fully implicit and nonlinearly consistent (INC) flow solver has been developed for the Navier-Stokes equation set. In our INC method, the equation set is solved by use of the Jacobian-free Newton-Krylov (JFNK) method. An efficient preconditioner has been developed which uses the semi-implicit method to solve the governing equations. Being that this preconditioner was designed to attack the fastest waves in the system and not other features in the implicit system such as advection or turbulent diffusion, the preconditioning technique is labeled as a physics-based preconditioner. A variety of linear solvers including SSOR, Krylov methods and/or multigrid approaches are used to approximately invert the pressure matrix in the semi-implicit algorithm. A suite of simulations will be conducted utilizing different linear solvers for the simple problem of the bouyant rise of a warm bubble. The problem will also document the ability of the INC approach to achieve second order in time accuracy.

New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion

This note presents an extension of previous work on physics-based preconditioning of the non-equilibrium radiation diffusion equations. The new physics-based preconditioner presented in this manuscript is a minor modification to the operator-split preconditioner presented previously. Results show that the new preconditioner is more effective on test problems that are more nonlinear.

Jacobian---Free Newton---Krylov Methods for the Accurate Time Integration of Stiff Wave Systems

Stiff wave systems are systems which exhibit a slow dynamical time scale while possessing fast wave phenomena. The physical effects of this fast wave may be important to the system, but resolving the fast time scale may not be required. When simulating such phenomena one would like to use time steps on the order of the dynamical scale for time integration. Historically, Semi-Implicit (SI) methods have been developed to step over the stiff wave time scale in a stable fashion. However, SI methods require some linearization and time splitting, and both of these can produce additional time integration errors. In this paper, the concept of using SI methods as preconditioners to Jacobian–Free Newton–Krylov (JFNK) methods is developed. This algorithmic approach results in an implicitly balanced method (no linearization or time splitting). In this paper, we provide an overview of SI methods in a variety of applications, and a brief background on JFNK methods. We will present details of our new algorithmic approach. Finally, we provide an overview of results coming from problems in geophysical fluid dynamics (GFD) and magnetohydrodynamics (MHD).

On Preconditioning Newton--Krylov Methods in Solidifying Flow Applications

Solidifying flow equations can be used to model industrial metallurgical processes such as casting and welding, and material science applications such as crystal growth. These flow equations contain locally stiff nonlinearities at the moving phase-change interface. We are developing a three-dimensional parallel simulation tool for such problems using a Jacobian-free Newton--Krylov solver and unstructured finite volume methods. A segregated (distributed, block triangular) preconditioning strategy is being developed for the Newton--Krylov solver. In this preconditioning approach we are only required to approximately invert matrices coming from a single field variable, not matrices arising from a coupled system. Additionally, simple linearizations are used in constructing our preconditioning operators. The preconditioning strategy is presented along with the performance of the methods. We consider problems in phase-change heat transfer and the thermally driven incompressible Navier--Stokes equations separately. This is a required intermediate step toward developing a successful preconditioning strategy for the fully coupled physics problem.

A Fully Implicit, Second Order in Time, Simulation of a Nuclear Reactor Core

This paper will present a high fidelity solution algorithm for a model of a nuclear reactor core barrel. This model consists of a system of nine nonlinearly coupled partial differential equations. The coolant is modeled with the 1-D six-equation two-phase flow model of RELAP5. Nonlinear heat conduction is modeled with a single 2-D equation. The fission power comes from two 2-D equations for neutron diffusion and precursor concentration. The solution algorithm presented will be the physics-based preconditioned Jacobian-free Newton-Krylov (JFNK) method. In this approach all nine equations are discretized and then solved in a single nonlinear system. Newtons method is used to iterate the nonlinear system to convergence. The Krylov linear solution method is used to solve the matrices in the linear steps of the Newton iterations. The physics-based preconditioner provides an approximation to the solution of the linear system that accelerates the Krylov iterations. Results will be presented for two algorithms. The first algorithm will be the traditional approach used by RELAP5. Here the two-phase flow equations are solved separately from the nonlinear conduction and neutron diffusion. Because of this splitting of the physics, and the linearizations employed this method is first order accurate in time. A second algorithm will be the JFNK method solved second order in time accurate. Results will be presented which compare these two algorithms in terms of accuracy and efficiency.Copyright

Temporal accuracy of the nonequilibrium radiation diffusion equations employing a Saha ionization model

This manuscript presents an analysis of the temporal accuracy of two first-order in time and two second-order in time integration methods as applied to a coupled radiation diffusion/reaction system of equations. These methods are categorized by their temporal order of accuracy, whether the algorithm includes operator splitting, and whether the algorithm includes linearizations. Accuracy of the different methods on three different test problems is discussed. These test problems are not new to the literature, but the purpose here is to demonstrate that it is possible to maintain second-order time accuracy on a nontrivial coupled system while employing an operator-split and linearized method.

A Multigrid Newton-Krylov Solver for Non-linear Systems

We present a technique which solves systems of nonlinear equations. The technique couples two solution methods together, multigrid and Newton-Krylov, producing in a method which efficiently uses the strengths of each technique. A form of distributed relaxation multigrid is used to solve systems of scalar linear equations which are then combined to provide efficient preconditioners for the Newton-Krylov method. This new method can be viewed as an alternative to other nonlinear multigrid solvers. Results will be presented for a steady state fluid flow and transient heat conduction.

Preconditioning and Solver Optimization Ideas for Radiative Transfer

In this paper, radiative transfer and time-dependent transport of radiation energy in participating media are modeled using a first-order spherical harmonics method (P1 ) and radiation diffusion. Partial differential equations for P1 and radiation diffusion are discretized by a variational form of the equations using support operators. Choices made in the discretization result in a symmetric positive definite (SPD) system of linear equations. Modeling multidimensional domains with complex geometries requires a very large system of linear equations with 10s of millions of elements. The computational domain is decomposed into a large number of subdomains that are solved on separate processors resulting in a massively parallel application. The linear system of equations is solved with a preconditioned conjugate gradient method. Various preconditioning techniques are compared in this study. Simple preconditioning techniques include: diagonal scaling, Symmetric Successive Over Relaxation (SSOR), and block Jacobi with SSOR as the block solver. Also, a two-grid multigrid-V-cycle method with aggressive coarsening is explored for use in the problems presented. Results show that depending on the test problem, simple preconditioners are effective, but the more complicated preconditioners such as an algebraic multigrid or the geometric multigrid are most efficient, particularly for larger problems and longer simulations. Optimal preconditioning varies depending on the problem and on how the physical processes evolve in time. For the insitu preconditioning techniques—SSOR and block Jacobi—a fuzzy controller can determine the optimal reconditioning process. Discussions of the current knowledge-based controller, an optimization search algorithm, are presented. Discussions of how this search algorithm can be incorporated into the development of data-driven controller incorporating clustering and subsequent construction of the fuzzy model from partitions are also discussed.Copyright

A Hybrid Solution Method for the Two-Phase Fluid Flow Equations

This paper will present a hybrid solution algorithm for the two-phase flow equations coupled to wall heat conduction. The partial differential equations in the physical model are the same as in RELAP5. The hybrid solution algorithm couples two solution methods, the solution method currently employed by RELAP5 and an implicitly balanced solution method. The RELAP5 solution method provides a fast solution that is “close” to the correct solution. The implicitly balanced solution method provides an accurate solution that is very stable. The resulting hybrid solution method is both fast and accurate. Results will be presented that show the hybrid solution method is more accurate than the RELAP5 solution method for the same size time step. In addition, results will be presented that show when accuracy and CPU time are considered simultaneously that there are ranges when the hybrid solution algorithm is preferred over the RELAP5 solution method.Copyright