Vincent A. Mousseau

An implicit nonlinearly consistent method for the two-dimensional shallow-water equations with Coriolis force

Abstract An implicit and nonlinearly consistent (INC) solution technique is presented for the two-dimensional shallow-water equations. Since the method is implicit, and therefore unconditionally stable, time steps may be used that result in both gravity wave Courant–Friedrichs–Lewy (CFL) numbers and advection CFL numbers being larger than one. By nonlinearly consistent it is meant that all of the unknown variables appear at the same time level in the equations and are solved for simultaneously in an iterative manner (i.e., no splitting errors in time). The INC method is compared to a more traditional semi-implicit method for stepping over the gravity wave stability constraint. Results are presented that show that the second-order-in-time INC method can maintain a high level of accuracy if the dynamical timescale of the system is resolved by the time step. To investigate this difference in temporal integration accuracy between the nonlinearly consistent method and the semi-implicit method an approximate mo...

ACCURACY AND EFFICIENCY OF A COUPLED NEUTRONICS AND THERMAL HYDRAULICS MODEL

The accuracy requirements for modern nuclear reactor simulation are steadily increasing due to the cost and regulation of relevant experimental facilities. Because of the increase in the cost of experiments and the decrease in the cost of simulation, simulation will play a much larger role in the design and licensing of new nuclear reactors. Fortunately as the work load of simulation increases, there are better physics models, new numerical techniques, and more powerful computer hardware that will enable modern simulation codes to handle the larger workload. This manuscript will discuss a numerical method where the six equations of two-phase flow, the solid conduction equations, and the two equations that describe neutron diffusion and precursor concentration are solved together in a tightly coupled, nonlinear fashion for a simplified model of a nuclear reactor core. This approach has two important advantages. The first advantage is a higher level of accuracy. Because the equations are solved together in a single nonlinear system, the solution is more accurate than the traditional “operator split” approach where the two-phase flow equations are solved first, the heat conduction is solved second and the neutron diffusion is solved third, limiting the temporal accuracy to 1st order because the nonlinear coupling between the physics is handled explicitly. The second advantage of the method described in this manuscript is that the time step control in the fully implicit system can be based on the timescale of the solution rather than a stability-based time step restriction like the material Courant. Results are presented from a simulated control rod movement and a rod ejection that address temporal accuracy for the fully coupled solution and demonstrate how the fastest timescale of the problem can change between the state variables of neutronics, conduction and two-phase flow during the course of a transient.

Application of the Newton–Krylov Method to Geophysical Flows

Abstract An implicit nonlinear algorithm, the Newton–Krylov method, for the efficient and accurate simulation of the Navier–Stokes equations, is presented. This method is a combination of a nonlinear outer Newton-based iteration and a linear inner conjugate residual (Krylov) iteration but does not require the explicit formation of the Jacobian matrix. This is referred to here as Jacobian-free Newton–Krylov (JFNK). The mechanics of the method are quite simple and the method has been previously used to solve a variety of complex coupled nonlinear equations. Like most Krylov-based schemes, the key to the efficiency of the method is preconditioning. Details concerning how preconditioning is implemented into this algorithm will be illustrated in a simple one-dimensional shallow-water framework. Another important aspect of this work is examining the accuracy and efficiency of the Newton–Krylov method against an explicit method of averaging (MOA) approach. This will aid in the determination of regimes for which ...

A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Euler equations on arbitrary grids. By taking advantage of handily available and yet invaluable information, namely the derivatives, in the context of the discontinuous Galerkin methods, a polynomial solution of one degree higher is reconstructed using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The resulting RDG method can be regarded as an improvement of a recovery-based DG method, in the sense that it shares the same nice features, such as high accuracy and efficiency, and yet overcomes some of its shortcomings such as a lack of flexibility, compactness, and robustness. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.

Temporal accuracy of the nonequilibrium radiation diffusion equations applied to two-dimensional multimaterial simulations

Abstract A study of the temporal accuracy of a variety of first- and second-order time-integration methods applied to two-dimensional, multimaterial, nonequilibrium, radiation diffusion simulations is presented. These methods are categorized by their temporal order of accuracy, whether the algorithm includes operator splitting, and whether the algorithm includes linearizations. Results are presented that simultaneously measure accuracy and efficiency of the different methods on two different test problems. The two test problems are designed to represent an easy problem, where different approximations may be accurate, and a hard test problem that will stress the different solution algorithms. Results show the importance of being second-order accurate in time and the importance of time-step control.

Marker Redistancing/Level Set Method for High-Fidelity Implicit Interface Tracking

A hybrid of the front tracking (FT) and the level set (LS) methods is introduced, combining advantages and removing drawbacks of both methods. The kinematics of the interface is treated in a Lagrangian (FT) manner, by tracking markers placed at the interface. The markers are not connected—instead, the interface topology is resolved in an Eulerian (LS) framework, by wrapping a signed distance function around Lagrangian markers each time the markers move. For accuracy and efficiency, we have developed a high-order “anchoring” algorithm and an implicit PDE-based redistancing. We have demonstrated that the method is 3rd-order accurate in space, near the markers, and therefore 1st-order convergent in curvature; this is in contrast to traditional PDE-based reinitialization algorithms, which tend to slightly relocate the zero level set and can be shown to be nonconvergent in curvature. The implicit pseudo-time discretization of the redistancing equation is implemented within the Jacobian-free Newton-Krylov (JFNK) framework combined with ILU(k) preconditioning. Due to the LS localization, the bandwidth of the Jacobian matrix is nearly constant, and the ILU preconditioning scales as

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

\sim N\log(\sqrt{N})

A Comparative Study of the Harmonic and Arithmetic Averaging of Diffusion Coefficients for Non-linear Heat Conduction Problems

in two dimensions, which implies efficiency and good scalability of the overall algorithm. We have demonstrated that the steady-state solutions in pseudo-time can be achieved very efficiently, with

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

\approx10

Fully-Implicit Interface Tracking for All-Speed Multifluid Flows

iterations (