Jacobian-free Newton-Krylov Method Research Articles

Analysis of the self-consistent IMEX method for tightly coupled non-linear systems

We present a mathematical analysis for our self-consistent Implicit/Explicit (IMEX) method that we have introduced in Kadioglu and Knoll (2010, 2013), Kadioglu et al. (2009, 2010) and Kadioglu (2017). The self-consistent IMEX algorithm is designed to produce second order time convergent solutions to multi-physics and multiple time scale fluid dynamics problems. The algorithm is a combination of an explicit block that solves the non-stiff part and an implicit block that solves the stiff part of the problem. The explicit block is always solved inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-Free Newton Krylov (JFNK) method (Knoll and Keyes, 2004; Nourgaliev et al., 2010; Saad, 2003). In this way, there is a continuous interaction between the implicit and explicit blocks meaning that the improved solutions (in terms of the time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all the nonlinearities due to coupling of different time terms are converged. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reduction in time accuracy for certain types of problems that a classical IMEX method can suffer from. We note that the classic IMEX methods split the operators such a way that the implicit and explicit blocks are executed independent of each other, and this may lead to non-converged nonlinearities therefore time inaccuracies for certain models. We also note that the well-known Strang-Splitting operator split technique (Strang, 1968 ) can suffer from the above mentioned time order reduction for certain applications, even though the method itself formally a second order numerical procedure. In this study, we provide a mathematical analysis (modified equation analysis) that examines and compares the time convergence behavior of our self-consistent IMEX method versus the classic IMEX method. We provide computational results to verify our analysis and analytical findings. We also computationally compare our IMEX procedure to the Strang-Splitting method.

Benchmarking the Multidimensional Stellar Implicit Code MUSIC

We present the results of a numerical benchmark study for the MUlti-dimensional Stellar Implicit Code (MUSIC) based on widely applicable two- and three-dimensional compressible hydrodynamics problems relevant to stellar interiors. MUSIC is an implicit large eddy simulation code that uses implicit time integration, implemented as a Jacobian-free Newton Krylov method. A physics based preconditioning technique which can be adjusted to target varying physics is used to improve the performance of the solver. The problems used for this benchmark study include the Rayleigh-Taylor and Kelvin-Helmholtz instabilities, and the decay of the Taylor-Green vortex. Additionally we show a test of hydrostatic equilibrium, in a stellar environment which is dominated by radiative effects. In this setting the flexibility of the preconditioning technique is demonstrated. This work aims to bridge the gap between the hydrodynamic test problems typically used during development of numerical methods and the complex flows of stellar interiors. A series of multi-dimensional tests are performed and analysed. Each of these test cases is analysed with a simple, scalar diagnostic, with the aim of enabling direct code comparisons. As the tests performed do not have analytic solutions we verify MUSIC by comparing to established codes including ATHENA and the PENCIL code. MUSIC is able to both reproduce behaviour from established and widely-used codes as well as results expected from theoretical predictions. This benchmarking study concludes a series of papers describing the development of the MUSIC code and provides confidence in the future applications.

JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow

The Jacobian-free Newton-Krylov (JFNK) method with an efficient physics-based preconditioner is applied for the numerical solution of the one-dimensional drift-flux model with the closure constitutive equations. Additional closure correlations, including the flow pattern dependent heat transfer correlations and the flow pattern independent kinematic constitutive correlation, are used to close the governing equations of one-dimensional drift-flux model. The governing equations have been discretized using the first-order upwind method for spatial discretization and the fully implicit method for temporal discretization. An efficient physics-based preconditioner derived from the semi-implicit solution of governing equations is used in the JFNK method to improve the efficiency and numerical stability. The numerical verification and code validation have been performed for subcooled boiling two-phase flow in a vertical tube. By comparing with the other methods (JFNK method without preconditioner, the Broyden method and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method), the preconditioning JFNK method shows the robustness and a good computational efficiency. Moreover, by comparing the numerical simulation results with the experimental results, it is found that the JFNK method with a physics-based preconditioner shows the good accuracy for the numerical simulation for one-dimensional boiling two-phase flow.

A Second-Order IMEX Method for Multi-Phase Flow Problems

We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Newton Krylov (JFNK) method [Knoll, D. A. and Keyes, D. E. [2004] Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397; Saad, Y. [2003] Iterative Methods for Sparse Linear Systems (Siam); Kelley, C. T. [2003] Solving Nonlinear Equations with Newton’s Method (Siam)]. In our algorithm implementation, the explicit block is embedded in the implicit block in a way that it is always part of the nonlinear function evaluation. In this way, there exists a continuous interaction between the implicit and explicit algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all nonlinearities due to coupling of different time terms are converged with the desired numerical time accuracy. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reductions in time convergence that is quite common in certain types of nonlinearly coupled systems. We remark that an incompressible multi-phase flow model can be a highly nonlinearly coupled system with the involvement of very stiff surface tension source terms. These kinds of flow problems are difficult to tackle numerically. In other words, highly nonlinear surface terms may remain unconverged leading to time inaccuracies or time order reductions to the first order even though the overall numerical scheme is designed as high order (second-order or higher) [Sussman, M. and Ohta, M. [2009] A stable and efficient method for treating surface tension in incompressible two-phase flow, SIAM J. Sci. Comput. 31(4), 2447–2471; Zheng, W., Zhu, B., Kim, B. and Fedkiw, R. [2015] A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension, J. Comput. Phys. 280, 96–142]. These and few more issues are addressed in this paper. We have numerically tested our newly proposed scheme by solving several multi-phase flow settings such as an air bubble rising in water, a Rayleigh–Taylor instability problem that is initiated by placing a heavy fluid on top of a lighter one, and a droplet problem in which a water droplet hits the pool of water. Our numerical results show that we have achieved the second-order time accuracy without any order reductions. Moreover, the interfaces between the fluids are captured reasonably well.

A new numerical method for solution of boiling flow using combination of SIMPLE and Jacobian-free Newton-Krylov algorithms

In this paper, an efficient and stable numerical approach is developed in which a different combination of Jacobian-free Newton-Krylov (JFNK) and SIMPLE methods are introduced for solution of two-phase flow in both steady state and transient conditions. It is shown that, combination of Krylov subspace methods and SIMPLE type preconditioner give a fast convergence in the solution of the Drift Flux Model (DFM). It is demonstrated that the stability problem with SIMPLE approach, for two-phase flow problems, can be avoided by combining with JFNK. RELAP5 code and experimental data are considered for verification study. For all benchmarks, the proposed methods predictions are in good agreement with experiment and RELAP5 results. The accuracy of the proposed schemes were studied and compared with the classical SIMPLE method. It was found that the present approach to two-phase flow simulation predicts accurate results over a wide pressure range; whereas the classical SIMPLE algorithm underestimated the void fraction at low pressures. Finally, a typical power transient is considered to demonstrate how the problem fastest time scale can change during the course of a transient. The present assessment of the numerical method showed that the time step control is not based on a stability time step restriction, like the material Courant limit.

A Jacobian-free Edge-based Galerkin Formulation for Compressible Flows

A parallel formulation of a Jacobian-free all Mach numbers solver on unstructured hybrid meshes is proposed. The Finite Element formulation is edge-based with flow stabilization obtained with either AUSM+ -up or Roe scheme. The linear system is solved via a Jacobian-Free Newton-Krylov (JFNK) method with Lower-Upper Symmetric GaussSeidel (LU-SGS) used as matrix-free preconditioner. The traditional formulation of LU-SGS is enriched by including the contributions from viscous fluxes and boundary conditions. The accuracy and efficiency of the proposed approach are demonstrated over cases ranging from low to high Mach numbers: subsonic flow over the Trap Wing, transonic flow over the ONERA M6 wing, supersonic flow over a sphere, supersonic flow over a waverider and finally hypersonic flow over a sphere.

Development of a novel 1D coupled neutronics/thermal-hydraulics code and its verification on PWR rod ejection accident benchmark

This paper presents a new 1D Neutronics/Thermal-hydraulics code ATAC-1D based on the advanced Jacobian-Free Newton-Krylov (JFNK) method and the low dimensional equivalent strategy. Conventional operator-splitting (OS) strategies are used to maintain its accuracy with small time steps and linearization of the nonlinear problem, which leads to slow computation speed and linearization error. The JFNK method solves the troubles in the coupled neutronics/thermal-hydraulics problems mentioned above. Furthermore, a core-wide three dimension to one dimension equivalent method has been developed to provide variable few-group parameters. Finally, the performance of the coupled neutronics/thermal-hydraulics code ATAC-1D is studied by simulating four OECD/NEA CRP PWR rod ejection benchmark problems. The simulation results are compared to the reference ones, which proves that the developed 1D code has a good accuracy and practicability in nuclear reactor transient calculation.

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

We present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov- Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.

Application of Jacobian-free Newton–Krylov method in implicitly solving two-fluid six-equation two-phase flow problems: Implementation, validation and benchmark

This work represents a first-of-its-kind successful application to employ advanced numerical methods in solving realistic two-phase flow problems with two-fluid six-equation two-phase flow model. These advanced numerical methods include high-resolution spatial discretization scheme with staggered grids (high-order) fully implicit time integration schemes, and Jacobian-free Newton–Krylov (JFNK) method as the nonlinear solver. The computer code developed in this work has been extensively validated with existing experimental flow boiling data in vertical pipes and rod bundles, which cover wide ranges of experimental conditions, such as pressure, inlet mass flux, wall heat flux and exit void fraction. Additional code-to-code benchmark with the RELAP5-3D code further verifies the correct code implementation. The combined methods employed in this work exhibit strong robustness in solving two-phase flow problems even when phase appearance (boiling) and realistic discrete flow regimes are considered. Transitional flow regimes used in existing system analysis codes, normally introduced to overcome numerical difficulty, were completely removed in this work. This in turn provides the possibility to utilize more sophisticated flow regime maps in the future to further improve simulation accuracy.

Solving phase appearance/disappearance two-phase flow problems with high resolution staggered grid and fully implicit schemes by the Jacobian-free Newton–Krylov Method

The phase appearance/disappearance issue presents serious numerical challenges in two-phase flow simulations. Although many existing nuclear reactor safety analysis codes use different kinds of treatments for the phase appearance/disappearance problem, there are no fully consensual solutions. In this paper, a high-resolution spatial discretization scheme on staggered grid mesh and fully implicit time integration methods was used in two-phase flow simulations. Numerical treatments of the momentum equations were introduced for the disappearing phase without the need to use a cut-off value on void fraction. A discretized nonlinear system was solved using the Jacobian-free Newton Krylov (JFNK) method, which does not require expensive and error-prone derivation and implementation of analytical Jacobian matrix. These methods were tested with several two-phase flow problems with phase appearance/disappearance phenomena considered, such as a linear advection problem, an oscillating manometer problem, and a sedimentation problem. The combined methods demonstrated extremely robust and stable behaviors in solving the one-dimensional two-phase flow problems with phase appearance/disappearance. High-resolution spatial discretization and second-order fully implicit method also demonstrated their capabilities in significantly reducing numerical errors.

A Jacobian-free Newton-Krylov method for time-implicit multidimensional hydrodynamics

This work is a continuation of our efforts to develop an efficient implicit solver for multidimensional hydrodynamics for the purpose of studying important physical processes in stellar interiors, such as turbulent convection and overshooting. We present an implicit solver that results from the combination of a Jacobian-free Newton-Krylov method and a preconditioning technique tailored to the inviscid, compressible equations of stellar hydrodynamics. We assess the accuracy and performance of the solver for both 2D and 3D problems for Mach numbers down to 10-6. Although our applications concern flows in stellar interiors, the method can be applied to general advection and/or diffusion-dominated flows. The method presented in this paper opens up new avenues in 3D modeling of realistic stellar interiors allowing the study of important problems in stellar structure and evolution.

Implicitly solving phase appearance and disappearance problems using two-fluid six-equation model

Phase appearance and disappearance issue presents serious numerical challenges in two-phase flow simulations using the two-fluid six-equation model. Numerical challenges arise from the singular equation system when one phase is absent, as well as from the discontinuity in the solution space when one phase appears or disappears. In this work, a high-resolution spatial discretization scheme on staggered grids and fully implicit methods were applied for the simulation of two-phase flow problems using the two-fluid six-equation model. A Jacobian-free Newton-Krylov (JFNK) method was used to solve the discretized nonlinear problem. An improved numerical treatment was proposed and proved to be effective to handle the numerical challenges. The treatment scheme is conceptually simple, easy to implement, and does not require explicit truncations on solutions, which is essential to conserve mass and energy. Various types of phase appearance and disappearance problems relevant to thermal-hydraulics analysis have been investigated, including a sedimentation problem, an oscillating manometer problem, a non-condensable gas injection problem, a single-phase flow with heat addition problem and a subcooled flow boiling problem. Successful simulations of these problems demonstrate the capability and robustness of the proposed numerical methods and numerical treatments. Volume fraction of the absent phase can be calculated effectively as zero.

Jacobian-Free Newton-Krylov based on nodal expansion method for neutronic-thermal hydraulic coupling problem

The traditional fixed-point iteration method is typically used for neutronics/thermal-hydralics coupling problems in most nuclear safety analysis codes. But the fixed-point iteration method has a tendency to fail to be used in computing the coupling problems due to slow convergence rate in some cases and even no convergence, and thus resulting in a limited efficiency, especially for the tight-coupling and fast-transient problems. In addition, for the reactor thermal-hydraulic calculation, the traditional finite difference or volume method (FDM or FVM) is used. However, both FDM and FVM require fine mesh size to achieve the desired precision and thus also result in a limited efficiency for the large scale problems. In this paper, to ensure the accuracy, efficiency and convergence for large-scale and complicated coupling problems, the new methods-NEM_JFNK are successfully developed to simultaneously solve the neutronics/thermal-hydralics coupling problems by combining the advantage of the efficient coarse nodal expansion method (NEM) and Jacobian-Free Newton-Krylov method (JFNK). The NEM has been widely used in the reactor physics analysis due to its high efficiency and accuracy in the reactor physics analysis, and it has proved to be superior to FDM and FVM. To improve the efficiency and accuracy for the large scale problems, the NEM is first extended to thermal hydraulic problems from the reactor physics calculation. Then all the governing equations of the neutronics/thermal-hydralics coupling problems can be discretized by the NEM and all the variables can be solved on the coarse meshes so that the size of coupling problems is greatly reduced. To ensure the high accuracy for the coupling problems on the coarse meshes, the high-order coefficients in NEM are successfully transferred between the coupling terms by our research. After that, to ensure the convergence of complicated coupling problems, JFNK based on the NEMs needs to be developed. However, the researches of JFNK based on the NEM in reactor analysis are less and the existing JFNK methods are mostly based on FVM or FDM or the finite element method. In this paper, the NEM discretization equations are successfully integrated into the framework of JFNK through the special treatment and the NEM_JFNK with linear-based preconditioner named LP_NEM_JFNK is also successfully developed. In addition, to take advantage of the existing code and avoid the construction of residual formulations, the non-residual construction NEM_JFNK named NRC_NEM_JFNK is presented and the black-box coupling method is achieved by NRC_NEM_JFNK so that the existing codes only need the simple modification to achieve the combination of the NEM and JFNK. Numerical results show LP_NEM_JFNK and NRC_NEM_JFNK outperform traditional fixed-point iteration method in the sense of convergence rate and efficiency. Further studies are needed to extend the NEM_JFNK method to the multi-dimensional neutronic/thermal hydraulic coupling problems in the high temperature gas-cooled reactor.

A communication-avoiding implicit–explicit method for a free-surface ocean model

We examine a nonlinear elimination method for the free-surface ocean equations based on barotropic–baroclinic decomposition. The two dimensional scalar continuity equation is treated implicitly with a preconditioned Jacobian-free Newton–Krylov method (JFNK). The remaining three dimensional equations are subcycled explicitly within the JFNK residual evaluation with a method known as nonlinear elimination. In this approach, the memory footprint of the underlying Krylov vector is greatly reduced over that required by fully coupled implicit methods. The method is second-order accurate and scales algorithmically, with allowed timesteps much larger than fully explicit methods. Moreover, the hierarchical nature of the algorithm lends itself readily to emerging architectures. In particular, we introduce a communication staging strategy for the three dimensional explicit system that greatly reduces the communication costs of the algorithm and provides a key advantage as communication costs continue to dominate relative to floating point costs in emerging architectures.

Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with Jacobian-free Newton–Krylov method

This paper presents a numerical investigation on using the Jacobian-free Newton–Krylov (JFNK) method to solve the two-phase flow four-equation drift flux model with realistic constitutive correlations (‘closure models’). The drift flux model is based on Isshi and his collaborators’ work. Additional constitutive correlations for vertical channel flow, such as two-phase flow pressure drop, flow regime map, wall boiling and interfacial heat transfer models, were taken from the RELAP5-3D Code Manual and included to complete the model. The staggered grid finite volume method and fully implicit backward Euler method was used for the spatial discretization and time integration schemes, respectively. The Jacobian-free Newton–Krylov method shows no difficulty in solving the two-phase flow drift flux model with a discrete flow regime map. In addition to the Jacobian-free approach, the preconditioning matrix is obtained by using the default finite differencing method provided in the PETSc package, and consequently the labor-intensive implementation of complex analytical Jacobian matrix is avoided. Extensive and successful numerical verification and validation have been performed to prove the correct implementation of the models and methods. Code-to-code comparison with RELAP5-3D has further demonstrated the successful implementation of the drift flux model.

A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation

In this study, we demonstrate a fully implicit algorithm for the multi-species, multidimensional Rosenbluth–Fokker–Planck equation which is exactly mass-, momentum-, and energy-conserving, and which preserves positivity. Unlike most earlier studies, we base our development on the Rosenbluth (rather than Landau) form of the Fokker–Planck collision operator, which reduces complexity while allowing for an optimal fully implicit treatment. Our discrete conservation strategy employs nonlinear constraints that force the continuum symmetries of the collision operator to be satisfied upon discretization. We converge the resulting nonlinear system iteratively using Jacobian-free Newton–Krylov methods, effectively preconditioned with multigrid methods for efficiency. Single- and multi-species numerical examples demonstrate the advertised accuracy properties of the scheme, and the superior algorithmic performance of our approach. In particular, the discretization approach is numerically shown to be second-order accurate in time and velocity space and to exhibit manifestly positive entropy production. That is, H-theorem behavior is indicated for all the examples we have tested. The solution approach is demonstrated to scale optimally with respect to grid refinement (with CPU time growing linearly with the number of mesh points), and timestep (showing very weak dependence of CPU time with time-step size). As a result, the proposed algorithm delivers several orders-of-magnitude speedup vs. explicit algorithms.

Applications of high-resolution spatial discretization scheme and Jacobian-free Newton–Krylov method in two-phase flow problems

The majority of the existing reactor system analysis codes were developed using low-order numerical schemes in both space and time. In many nuclear thermal–hydraulics applications, it is desirable to use higher-order numerical schemes to reduce numerical errors. High-resolution spatial discretization schemes provide high order spatial accuracy in smooth regions and capture sharp spatial discontinuity without nonphysical spatial oscillations. In this work, we adapted an existing high-resolution spatial discretization scheme on staggered grids in two-phase flow applications. Fully implicit time integration schemes were also implemented to reduce numerical errors from operator-splitting types of time integration schemes. The resulting nonlinear system has been successfully solved using the Jacobian-free Newton–Krylov (JFNK) method. The high-resolution spatial discretization and high-order fully implicit time integration numerical schemes were tested and numerically verified for several two-phase test problems, including a two-phase advection problem, a two-phase advection with phase appearance/disappearance problem, and the water faucet problem. Numerical results clearly demonstrated the advantages of using such high-resolution spatial and high-order temporal numerical schemes to significantly reduce numerical diffusion and therefore improve accuracy. Our study also demonstrated that the JFNK method is stable and robust in solving two-phase flow problems, even when phase appearance/disappearance exists.

Jacobian-Free Newton Krylov based iterative strategies in frequency-domain analyses

This study examines the theory and performance of Jacobian-Free Newton Krylov (JFNK) methods for the efficient, iterative solution of steady state dynamics of linear vibrating systems in the frequency domain. Currently, most commercial FEM algorithms employ use direct factorization of large system matrices to achieve steady state solution. Such approaches are usually quite demanding on memory and CPU requirements. Some implementations exploit iterative solutions, but still use large, assembled system matrices, requiring significant memory. The methods investigated in this work avoid the formation of a matrix completely, minimizing memory requirements and enabling much larger problems to be performed on desktop computers. Here, the Conjugate-Gradient (CG) and Transpose Free Quasi Minimal Residual (TFQMR) algorithms are studied as possibilities. These methods are of particular interest for adaptation to finite element software which uses explicit transient dynamics, because such software's optimal architecture prevents the formation and solution of a stiffness (or a tangent stiffness) matrix. In order to demonstrate the advantages of these algorithms, we choose examples that use the JFNK-CG and JFNK-TFQMR technique to show computational advantages over regular matrix based solutions, with significant decrease in memory requirements.

Fully Implicit Ultrascale Physics Solvers and Application to Ion Source Modeling

Many problems of interest in plasma modeling are subject to the tyranny of scales, specifically, problems that encompass physical processes that operate on timescales that are separated by many orders of magnitude. Investigating such problems, therefore, requires the use of implicit time-integration schemes, which advance problem solutions on the timescale of interest, while incorporating the physics of the fast timescales. One promising route to develop these implicit solvers is the combination of Jacobian-free Newton–Krylov (JFNK) methods, but adapting these methods to work in ultrascale computing environments is a formidable challenge. Here, we describe research on new approaches to adapt algebraic mulgrid-based solvers (that can be used for providing efficient preconditioners for JFNK methods) to ultrascale computing environments, the development and testing of JFNK solvers for coupled plasma electromagnetics within the USIM framework and the application of these methods to modeling H− ion sources for the spallation neutron source at ORNL.

A parallel multi-domain solution methodology applied to nonlinear thermal transport problems in nuclear fuel pins

This paper describes an efficient and nonlinearly consistent parallel solution methodology for solving coupled nonlinear thermal transport problems that occur in nuclear reactor applications over hundreds of individual 3D physical subdomains. Efficiency is obtained by leveraging knowledge of the physical domains, the physics on individual domains, and the couplings between them for preconditioning within a Jacobian Free Newton Krylov method. Details of the computational infrastructure that enabled this work, namely the open source Advanced Multi-Physics (AMP) package developed by the authors is described. Details of verification and validation experiments, and parallel performance analysis in weak and strong scaling studies demonstrating the achieved efficiency of the algorithm are presented. Furthermore, numerical experiments demonstrate that the preconditioner developed is independent of the number of fuel subdomains in a fuel rod, which is particularly important when simulating different types of fuel rods. Finally, we demonstrate the power of the coupling methodology by considering problems with couplings between surface and volume physics and coupling of nonlinear thermal transport in fuel rods to an external radiation transport code.