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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 (21 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 178 (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 156 (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...
Accurate Projection Methods for the Incompressible NavierStokes Equations
, 2001
"... This paper considers the accuracy of projection method approximations to the ..."
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Cited by 132 (6 self)
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This paper considers the accuracy of projection method approximations to the
An immersed interface method for incompressible navierstokes equations
 SIAM J. Sci. Comput
, 2003
"... Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discr ..."
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Cited by 99 (3 self)
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Abstract. The method developed in this paper is motivated by Peskin’s immersed boundary (IB) method, and allows one to model the motion of flexible membranes or other structures immersed in viscous incompressible fluid using a fluid solver on a fixed Cartesian grid. The IB method uses a set of discrete delta functions to spread the entire singular force exerted by the immersed boundary to the nearby fluid grid points. Our method instead incorporates part of this force into jump conditions for the pressure, avoiding discrete dipole terms that adversely affect the accuracy near the immersed boundary. This has been implemented for the twodimensional incompressible Navier–Stokes equations using a highresolution finitevolume method for the advective terms and a projection method to enforce incompressibility. In the projection step, the correct jump in pressure is imposed in the course of solving the Poisson problem. This gives sharp resolution of the pressure across the interface and also gives better volume conservation than the traditional IB method. Comparisons between this method and the IB method are presented for several test problems. Numerical studies of the convergence and order of accuracy are included.
The immersed interface method for the Navier–Stokes equations with singular forces
 J. Comput. Phys
"... Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, t ..."
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Cited by 83 (5 self)
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Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be firstorder accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a secondorder projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully secondorder accuracy for the velocity and nearly secondorder accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. c ○ 2001 Academic Press Key Words: Navier–Stokes equations; interface; discontinuous and nonsmooth solution; immersed interface method; immersed boundary method; projection method; level set method. 1.
On error estimates of projection methods for NavierStokes equations: First order schemes
 SIAM J. Numer. Anal
, 1992
"... Abstract. We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible NavierStokes equations. The error analysis is accomplished by interpreting the respective projection schemes as secondorder time discretizations of a pert ..."
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Cited by 74 (13 self)
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Abstract. We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible NavierStokes equations. The error analysis is accomplished by interpreting the respective projection schemes as secondorder time discretizations of a perturbed system which approximates the NavierStokes equations. Numerical results in agreement with the error analysis are also presented. 1.
A Stochastic Projection Method for Fluid Flow I. Basic Formulation
 J. Comput. Phys
, 2001
"... We describe the construction and implementation of a stochastic NavierStokes... ..."
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Cited by 67 (3 self)
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We describe the construction and implementation of a stochastic NavierStokes...
The immersed boundary method: a projection approach.
 J. Comput. Phys.,
, 2007
"... Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface t ..."
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Cited by 59 (12 self)
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Abstract A new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion. Like previous methods, a boundary force is applied at the immersed surface to satisfy the noslip constraint. This extra constraint can be added to the incompressible NavierStokes equations by introducing regularization and interpolation operators. The current method gives prominence to the role of the boundary force acting as a Lagrange multiplier to satisfy the noslip condition. This role is analogous to the effect of pressure on the momentum equation to satisfy the divergencefree constraint. The current immersed boundary method removes slip and nondivergencefree components of the velocity field through a projection. The boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger CFL numbers compared to some past methods. Symmetry and positivedefiniteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field. Examples show that the current formulation achieves secondorder temporal accuracy and better than firstorder spatial accuracy in L 2 norms for oneand twodimensional test problems. Results from twodimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies.