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. Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be approximated in this manner. In this work we systematically analyze the performance of such schemes, propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods. For the prototype linear advection-diffusion equation, a stability analysis for first, second, third and fourth order multistep IMEX schemes is performed. Stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified. Numerical experiments demonstrate that weak decay of high freque...

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have built-in machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible

During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^-1-compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools -- the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finite-difference schemes; error estimates derived from the one-sided stability of Godunov-type methods for convex conservation laws (and their multidimensional analogue -- viscosity solutions of demi-concave Hamilton-Jacobi equations); we outline, in the one-dimensional case, the convergence proof of finite-element streamline-diffusion and spectral viscosity schemes based on the div-curl lemma; we also address the questions of convergence and error estimates for multidimensional finite-volume schemes on non-rectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finite-volume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.

Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance. Key words. least-squares finite element methods, elliptic equations AMS subject classifications. 65N30 1. Introduction. The

Two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one- and two-dimensional hydrodynamic and magnetohydrodynamic problems. Two of the schemes, YDFCT and TVDLF are tested extensively for the first time. The results give an insight into the limitations of the methods, their relative strengths and weaknesses. Some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised. 1 INTRODUCTION Many interesting and important problems arise in astrophysical, solar, magnetospheric and thermonuclear research which can be described by the system of magnetohydrodynamic (MHD) equations. The complexity of these problems often prohibits an analytical investigation and/or only some of the variables can be observed or measured experimentally, thus the researcher has to rely on numerical simulations. In many situations, MHD flows develop steep gradients, shock waves, contact disconti...

Abstract. In this paper we present a Krylov acceleration technique for nonlinear PDEs. As a ‘preconditioner’ we use nonlinear multigrid schemes such as the Full Approximation Scheme (FAS) [1]. The benefits of nonlinear multigrid used in combination with the new accelerator are illustrated by difficult nonlinear elliptic scalar problems, such as the Bratu problem, and for systems of nonlinear equations, such as the Navier-Stokes equations. Key words. nonlinear Krylov acceleration, nonlinear multigrid, robustness, restarting conditions. AMS subject classifications. 65N55, 65H10, 65Bxx. 1. Introduction. It is well known that multigrid solution methods are optimal O(N) solvers, when all components in a method are chosen correctly. For difficult problems, such as some systems of nonlinear equations, it is far from trivial to choose these optimal components. The influence on the multigrid convergence of combinations of complicated factors, like convection-dominance, anisotropies, nonlinearities or non M-matrix properties (the

The Parallel Diagonal Dominant (PDD) algorithm is an efficient tridiagonal solver. In this paper, a detailed study of the PDD algorithm is given. First the PDD algorithm is extended to solve periodic tridiagonal systems and its scalability is studied. Then the reduced PDD algorithm, which has a smaller operation count than that of the conventional sequential algorithm for many applications, is proposed. Accuracy analysis is provided for a class of tridiagonal systems, the symmetric and skew-symmetric Toeplitz tridiagonal systems. Implementation results show that the analysis gives a good bound on the relative error, and the PDD and reduced PDD algorithms are good candidates for emerging massively parallel machines. Index Terms: Parallel processing, Parallel numerical algorithms, Scalable computing, Tridiagonal systems, Toeplitz systems Manuscript received April 7, 1993; revised April 7, 1994 and January 27, 1995. This research was supported in part by the National Aeronautics and S...

Abstract. High-accuracy finite-difference schemes are used to solve the two-dimensional timedomain Maxwell equations for electromagnetic wave propagation and scattering. The high-accuracy schemes consist of a seven-point spatial operator coupled with a six-stage Runge–Kutta time-marching method. Two methods are studied, one of which produces the maximum order of accuracy and one of which is optimized for propagation distances smaller than roughly 300 wavelengths. Boundary conditions are presented which preserve the accuracy of these schemes when modeling interfaces between different materials. Numerical experiments are performed which demonstrate the utility of the high-accuracy schemes in modeling waves incident on dielectric and perfect-conducting scatterers using Cartesian and curvilinear grids. The high-accuracy schemes are shown to be substantially more efficient, in both computing time and memory, than a second-order and a fourth-order method. The optimized scheme can lead to a reduction in error relative to the maximum-order scheme, with no additional expense, especially when the number of wavelengths of travel is large.