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217
Functional Phonology  Formalizing the interactions between articulatory and perceptual drives
, 1998
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ImplicitExplicit Methods For TimeDependent PDEs
 SIAM J. NUMER. ANAL
, 1997
"... Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection ..."
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Cited by 177 (6 self)
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Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reactiondiffusion 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 advectiondiffusion 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...
Approximate Solutions of Nonlinear Conservation Laws and Related Equations
, 1997
"... 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 ..."
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Cited by 55 (12 self)
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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^1compact 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) finitedifference schemes; error estimates derived from the onesided stability of Godunovtype methods for convex conservation laws (and their multidimensional analogue  viscosity solutions of demiconcave HamiltonJacobi equations); we outline, in the onedimensional case, the convergence proof of finiteelement streamlinediffusion and spectral viscosity schemes based on the divcurl lemma; we also address the questions of convergence and error estimates for multidimensional finitevolume schemes on nonrectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finitevolume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... 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 loc ..."
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Cited by 53 (18 self)
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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 builtin 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
Parallel NewtonKrylovSchwarz Algorithms For The Transonic Full Potential Equation
, 1998
"... We study parallel twolevel 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, NewtonKrylovSchwarz (NKS), employs an inexact finite ..."
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Cited by 51 (32 self)
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We study parallel twolevel 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, NewtonKrylovSchwarz (NKS), employs an inexact finitedifference Newton method and a Krylov space iterative method, with a twolevel 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 elliptichyperbolic 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 fillin in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributedmemory parallel computer.
Finite element methods of leastsquares type
, 1998
"... We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticit ..."
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Cited by 51 (3 self)
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We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convectiondiffusion. For many of these problems, leastsquares 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 leastsquares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the infsup condition arising in mixed methods for the Stokes and NavierStokes equations. As a result, application of leastsquares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.
Comparison of some Flux Corrected Transport and Total Variation Diminishing Numerical Schemes for Hydrodynamic and Magnetohydrodynamic Problems
, 1996
"... Two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and twodimensional hydrodynamic and magnetohydrodynamic problems. Two of the schemes, YDFCT and TVDLF are tested extensively for the first time. The results give an insight i ..."
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Cited by 43 (7 self)
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Two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and twodimensional 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...
Parallel scientific computing in C++ and MPI: a seamless approach to parallel algorithms and their implementation
, 2003
"... Scientific computing is by its very nature a practical subject it requires tools and a lot of practice. To solve realistic problems we need not only fast algorithms but also a combination of good tools and fast computers. This is the subject of the current book, which emphasizes equally all three: ..."
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Cited by 31 (2 self)
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Scientific computing is by its very nature a practical subject it requires tools and a lot of practice. To solve realistic problems we need not only fast algorithms but also a combination of good tools and fast computers. This is the subject of the current book, which emphasizes equally all three: algorithms, tools, and computers. Often times such concepts and tools are taught serially across different courses and different textbooks, and hence the interconnection between them is not immediately apparent. We believe that such a close integration is important from the outset. The book starts with a heavy dosage of C++ and basic mathematical and computational concepts, and it ends emphasizing advanced parallel algorithms that are used in modern simulations. We have tried to make this book fun to read, to somewhat demystify the subject, and thus the style is sometimes informal and personal. It may seem that this happens at the expense of rigor, and indeed we have tried to limit notation and theorem proofing. Instead, we emphasize concepts and useful tricksofthetrade with many code segments, remarks, reminders, and warnings throughout the book. The material of this book has been taught at different times to students in engineering,
A Nonlinear Additive Schwarz Preconditioned Inexact Newton Method for Shocked Duct Flow
 Proceedings of the 13th International Conference on Domain Decomposition Methods
, 2001
"... this paper, we shall use the nonlinear additive Schwarz algorithm as the preconditioner and focus on the performance of PIN for a compressible shock tube problem, which is known to be a dicult test case for inexact Newton type algorithms. ..."
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Cited by 29 (14 self)
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this paper, we shall use the nonlinear additive Schwarz algorithm as the preconditioner and focus on the performance of PIN for a compressible shock tube problem, which is known to be a dicult test case for inexact Newton type algorithms.
Highresolution FEMFCT schemes for multidimensional conservation laws
"... The fluxcorrectedtransport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory loworder method of upwind type is derived by elimination of negative offdiagonal entries of the discrete transport ope ..."
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Cited by 29 (16 self)
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The fluxcorrectedtransport paradigm is generalized to implicit finite element schemes and nonlinear systems of hyperbolic conservation laws. In the scalar case, a nonoscillatory loworder method of upwind type is derived by elimination of negative offdiagonal entries of the discrete transport operator. The difference between the discretizations of high and low order is decomposed into a sum of skewsymmetric antidiffusive fluxes. An iterative flux limiter independent of the time step is proposed for implicit schemes. The nonlinear antidiffusion is incorporated into the solution in the framework of a defect correction scheme preconditioned by the monotone loworder operator. In the case of a hyperbolic system, the global Jacobian matrix is assembled edgebyedge without resorting to numerical integration. Its loworder counterpart is constructed by rendering all offdiagonal blocks positive definite or adding scalar artificial diffusion proportional to the spectral radius of the Roe matrix. The coupled equations are solved in a segregated manner within an outer defect correction loop equipped with a blockdiagonal preconditioner. After a suitable synchronization, the correction factors evaluated for an arbitrary set of indicator variables are applied to the antidiffusive fluxes which are inserted into the global defect vector. The performance of the new algorithm is illustrated by numerical examples for scalar transport problems and the compressible Euler equations. Key Words: convectiondominated flows; hyperbolic conservation laws; flux correction; finite elements; implicit timestepping 1