Abstract. We introduce a well-developed Newton iterative (truncated Newton) algorithm for solving large-scale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and features. The structure offers the user great flexibility in addressing problem specificity through preconditioning and other means and allows easy adaptation to parallel environments. Features and capabilities are illustrated in numerical experiments.

Globalized inexact Newton methods are well suited for solving large-scale systems of nonlinear equations. When combined with a Krylov iterative method, an explicit Jacobian is never needed, and the resulting matrix-free Newton-Krylov method greatly simplifies application of the method to complex problems. Despite asymptotically superlinear rates of convergence, the overall efficiency of a Newton-Krylov solver is determined by the preconditioner. High-quality preconditioners can be constructed from methods that incorporate problem-specific information, and for the incompressible Navier-Stokes equations, classical pressure-correction methods such as SIMPLE and SIMPLER fulfill this requirement. A preconditioner is constructed by using these pressurecorrection methods as smoothers in a linear multigrid procedure. The effectiveness of the resulting Newton-Krylov-multigrid method is demonstrated on benchmark incompressible flow problems.

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.

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Diffusion approximations to radiation transport feature a nonlinear conduction coefficient that leads to formation of a sharp front, or Marshak wave, under suitable initial and boundary conditions. The front can vary several orders of magnitude over a very short distance. Resolving the shape of the Marshak wave is essential, but using a global fine mesh can be prohibitively expensive. In such circumstances it is natural to consider using adaptive mesh refinement (AMR) to place a fine mesh only in the vicinity of the propagating front. In addition, to avoid any loss of accuracy due to linearization, implicit time integration should be used to solve the equilibrium radiation diffusion equation. Implicit time integration on AMR grids introduces a new challenge, as algorithmic complexity must be controlled to fully realize the performance benefits of AMR. A Newton-Krylov method together with a multigrid preconditioner addresses this latter issue on a uniform grid. A straightforward generalization is to use a multilevel preconditioner that is tuned to the structure of the AMR grid, such as the fast adaptive composite grid (FAC) method. We describe the resulting Newton-Krylov-FAC method and demonstrate its performance on simple equilibrium radiation diffusion problems. Introduction.Radiation transport plays an important role in numerous fields of study, including astrophysics, laser fusion, combustion applications, atmospheric dynamics, and medical imaging. When photon mean free paths are much shorter than characteristic length scales, a diffusion approximation provides a reasonably accurate description of radiation penetrating from a hot source to a cold medium. This approximation features a nonlinear conduction coefficient that leads to formation of a sharp front, in which the solution can vary several orders of magnitude over a very short distance. The shape of the front can be very complex as it interacts with different materials having different conduction properties. Resolving these localized features with a global fine mesh can be prohibitively expensive. It is natural to consider reducing the cost of accurately resolving these fronts by using adaptive mesh refinement (AMR), which concentrates computational effort by increasing spatial resolution only locally.Classical solution techniques for equilibrium radiation diffusion use a linearized conduction coefficient to avoid the expense of solving a system of nonlinear equations at each time step. This introduces a first-order error in time that precludes effective use of higher-order time integration methods, and requires small time steps to maintain time accuracy. Analytic and computational results that demonstrate degradation in time accuracy associated with linearization in the presence of strong nonlinear coefficients can be found in [24,15]. Such effects can be avoided by using im-