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An overview of projection methods for incompressible flows
 Comput. Methods Appl. Mech. Engrg
"... Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1no ..."
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Cited by 203 (20 self)
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Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1norm for the velocity and of the L 2norm for the pressure. We also show that the class of fractionalstep methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocitycorrection methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods. Key words. Navier–Stokes equations, projection methods, fractionalstep methods, incompressibility, finite elements, spectral approximations
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...
ImplicitExplicit RungeKutta Methods for TimeDependent Partial Differential Equations
 Appl. Numer. Math
, 1997
"... Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an ap ..."
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Cited by 157 (7 self)
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Implicitexplicit (IMEX) linear multistep timediscretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable timestep restrictions when applied to convectiondiffusion problems, unless diffusion strongly dominates and an appropriate BDFbased scheme is selected [2]. In this paper, we develop RungeKuttabased IMEX schemes that have better stability regions than the best known IMEX multistep schemes over a wide parameter range. 1 Introduction When a timedependent partial differential equation (PDE) involves terms of different types, it is a natural idea to employ different discretizations for them. Implicitexplicit (IMEX) timediscretization schemes are an example of such a strategy. Linear multistep IMEX schemes have been used by many researchers, especially in conjunction with spectral methods [10, 3]. Some schemes of this type were proposed and analyzed as far back as the late 1970's [15, 5]. Instances of...
A MATLAB differentiation matrix suite
 ACM TOMS
, 2000
"... A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolan ..."
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Cited by 68 (3 self)
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A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.
Fast Numerical Methods for Stochastic Computations: A Review
, 2009
"... This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework ..."
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Cited by 65 (2 self)
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This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multidimensional random spaces.
Finite element heterogeneous multiscale methods with near optimal . . .
"... This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro s ..."
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Cited by 58 (24 self)
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This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro solver. Unlike the micromacro methods based on standard FEM proposed so far in HMM we obtain, for periodic homogenization problems, a method that has almostlinear complexity in the number of degrees of freedom of the discretization of the macro (slow) variable.
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 45 (16 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Missing point estimation in models described by proper orthogonal decomposition
, 2007
"... This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used w ..."
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Cited by 42 (7 self)
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This paper presents a new method of Missing Point Estimation (MPE) to derive efficient reducedorder models for largescale parametervarying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projectionbased model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of datadependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25 % of the spatial grid points without compromising the accuracy of the reduced model.
LaguerreGalerkin method for nonlinear partial differential equations on a semiinfinite interval
, 2000
"... A LaguerreGalerkin method is proposed and analyzed for the Burgers equation and BenjaminBonaMahony (BBM) equation on a semiinfinite interval. By reformulating these equations with suitable functional transforms, it is shown that the LaguerreGalerkin approximations are convergent on a semiinfini ..."
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Cited by 37 (18 self)
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A LaguerreGalerkin method is proposed and analyzed for the Burgers equation and BenjaminBonaMahony (BBM) equation on a semiinfinite interval. By reformulating these equations with suitable functional transforms, it is shown that the LaguerreGalerkin approximations are convergent on a semiinfinite interval withspectral accuracy. An efficient and accurate algorithm based on the LaguerreGalerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.