Carol S. Woodward

SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers

SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

Multiphysics simulations: Challenges and opportunities

We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

Development of a Coupled Groundwater–Atmosphere Model

Abstract Complete models of the hydrologic cycle have gained recent attention as research has shown interdependence between the coupled land and energy balance of the subsurface, land surface, and lower atmosphere. PF.WRF is a new model that is a combination of the Weather Research and Forecasting (WRF) atmospheric model and a parallel hydrology model (ParFlow) that fully integrates three-dimensional, variably saturated subsurface flow with overland flow. These models are coupled in an explicit, operator-splitting manner via the Noah land surface model (LSM). Here, the coupled model formulation and equations are presented and a balance of water between the subsurface, land surface, and atmosphere is verified. The improvement in important physical processes afforded by the coupled model using a number of semi-idealized simulations over the Little Washita watershed in the southern Great Plains is demonstrated. These simulations are initialized with a set of offline spinups to achieve a balanced state of ini...

A parallel, implicit, cell‐centered method for two‐phase flow with a preconditioned Newton–Krylov solver

A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective.

Preconditioning Strategies for Fully Implicit Radiation Diffusion with Material-Energy Transfer

In this paper, we present a comparison of four preconditioning strategies for Jacobian systems arising in the fully implicit solution of radiation diffusion coupled with material energy transfer. The four preconditioning methods are block Jacobi, Schur complement, and operator splitting approaches that split the preconditioner solve into two steps. One splitting method includes the coupling of the radiation and material fields that appears in the matrix diagonal in the first solve, and the other method puts this coupling into the second solve. All preconditioning approaches use multigrid methods to invert blocks of the matrix formed from the diffusion operator. The Schur complement approach is clearly seen to be the most effective for a large range of weightings between the diffusion and energy coupling terms. In addition, tabulated opacity studies were conducted where, again, the Schur preconditioner performed well. Last, a parallel scaling study was done showing algorithmic scalability of the Schur preconditioner.

A fully implicit numerical method for single-fluid resistive magnetohydrodynamics

We present a nonlinearly implicit, conservative numerical method for integration of the single-fluid resistive MHD equations. The method uses a high-order spatial discretization that preserves the solenoidal property of the magnetic field. The fully coupled PDE system is solved implicitly in time, providing for increased interaction between physical processes as well as additional stability over explicit-time methods. A high-order adaptive time integration is employed, which in many cases enables time steps ranging from one to two orders of magnitude larger than those constrained by the explicit CFL condition. We apply the solution method to illustrative examples relevant to stiff magnetic fusion processes which challenge the efficiency of explicit methods. We provide computational evidence showing that for such problems the method is comparably accurate with explicit-time simulations, while providing a significant runtime improvement due to its increased temporal stability.

Implicit solvers for large-scale nonlinear problems

Computational scientists are grappling with increasingly complex, multi-rate applications that couple such physical phenomena as fluid dynamics, electromagnetics, radiation transport, chemical and nuclear reactions, and wave and material propagation in inhomogeneous media. Parallel computers with large storage capacities are paving the way for high-resolution simulations of coupled problems; however, hardware improvements alone will not prove enough to enable simulations based on brute-force algorithmic approaches. To accurately capture nonlinear couplings between dynamically relevant phenomena, often while stepping over rapid adjustments to quasi-equilibria, simulation scientists are increasingly turning to implicit formulations that require a discrete nonlinear system to be solved for each time step or steady state solution. Recent advances in iterative methods have made fully implicit formulations a viable option for solution of these large-scale problems. In this paper, we overview one of the most effective iterative methods, Newton-Krylov, for nonlinear systems and point to software packages with its implementation. We illustrate the method with an example from magnetically confined plasma fusion and briefly survey other areas in which implicit methods have bestowed important advantages, such as allowing high-order temporal integration and providing a pathway to sensitivity analyses and optimization. Lastly, we overview algorithm extensions under development motivated by current SciDAC applications.

On Mesh-Independent Convergence of an Inexact Newton--Multigrid Algorithm

In this paper we revisit and prove optimal-order and mesh-independent convergence of an inexact Newton method where the linear Jacobian systems are solved with multigrid techniques. This convergence is shown using Banach spaces and the norm

A federated simulation toolkit for electric power grid and communication network co-simulation

\max\{\|\cdot\|_1,\; \|\cdot\|_{0,\infty}\}

Operator-Based Preconditioning of Stiff Hyperbolic Systems

, a stronger norm than is used in previous work. These results are valid for a class of second-order, semilinear, finite element, elliptic problems posed on quasi-uniform grids. Numerical results are given which validate the theory.