A pure-compact scheme for the streamfunction formulation of Navier–Stokes equations

Ben‐Artzi, Matania; Croisille, Jean‐Pierre; Fishelov, Dalia; Trachtenberg, Shlomo

doi:10.1016/j.jcp.2004.11.024

Cited by 70 publications

(86 citation statements)

References 20 publications

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“…In this paper we continue the study, which was initiated in [10,28,9,7], of the numerical resolution of the pure streamfunction formulation of the timedependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in [28,9,7], to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators.…”

mentioning

confidence: 93%

“…We mention also [48,20,40,39,34] for works on the stationary Stokes or Navier-Stokes equation. In [9,7] a comprehensive treatment of a second order compact scheme in space and time is presented. It is based on the Stephenson scheme for the biharmonic problem [50] and includes a detailed analysis of the (linearized) stability and a proof of the convergence of the fully nonlinear scheme.…”

Section: Introductionmentioning

confidence: 99%

“…We note also that a compact finite-difference (second-order) scheme, based on the same approach, for irregular domains , has recently been presented [6]. Recall that an important feature of the methodology presented in [9], [7] is that the "numerical boundary conditions" are applied only to the streamfunction itself and imposed solely on the boundary. Thus the scheme conforms exactly with the theoretical (pure streamfunction) formulation of the Navier-Stokes system .…”

Section: Introductionmentioning

confidence: 99%

“…The main purpose of the present paper is to extend the aforementioned second order scheme [9], to a fourth order scheme. With this added accuracy, we are able to simulate the dynamics of flow problems in rectangles with sparser grids and fewer time steps, compared with the second order scheme.…”

Section: Introductionmentioning

confidence: 99%

“…The first is a second order time-stepping scheme, already used in [9]. The second is formally almost third order accurate and was introduced in [49] in the context of Navier-Stokes simulations using spectral methods for the discretization in space.…”

Section: Introductionmentioning

confidence: 99%

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A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

Ben‐Artzi

Croisille²,

Fishelov

2009

J Sci Comput

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Abstract. In this paper we continue the study, which was initiated in [10,28,9,7], of the numerical resolution of the pure streamfunction formulation of the timedependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in [28,9,7], to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is actually obtained for several testcases.

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mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

Ben‐Artzi

Croisille²,

Fishelov

2009

J Sci Comput

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A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications

Wong

Chan

2005

Numerical Methods in Fluids

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SUMMARYA well-recognized approach for handling the incompressibility constraint by operating directly on the discretized Navier-Stokes equations is used to obtain the decoupling of the pressure from the velocity ÿeld. By following the current developments by Guermond and Shen, the possibilities of obtaining accurate pressure and reducing boundary-layer e ect for the pressure are analysed. The present study mainly reports the numerical solutions of an unsteady Navier-Stokes problem based on the so-called consistent splitting scheme (J. Comput. Phys. 2003; 192:262-276). At the same time the Dirichlet boundary value conditions are considered. The accuracy of the method is carefully examined against the exact solution for an unsteady ow physics problem in a simply connected domain. The e ectiveness is illustrated viz. several computations of 2D double lid-driven cavity problems.

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The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

Kalita¹,

Sen²

2007

Numerical Methods in Fluids

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SUMMARYWe recently proposed an improved (9,5) higher order compact (HOC) scheme for the unsteady twodimensional (2-D) convection-diffusion equations. Because of using only five points at the current time level in the discretization procedure, the scheme was seen to be computationally more efficient than its predecessors. It was also seen to capture very accurately the solution of the unsteady 2-D Navier-Stokes (N-S) equations for incompressible viscous flows in the stream function-vorticity ( − ) formulation.In this paper, we extend the scope of the scheme for solving the unsteady incompressible N-S equations based on primitive variable formulation on a collocated grid. The parabolic momentum equations are solved for the velocity field by a time-marching strategy and the pressure is obtained by discretizing the elliptic pressure Poisson equation by the steady-state form of the (9,5) scheme with the Neumann boundary conditions. In particular, for pressure, we adopt a strategy on the collocated grid in conjunction with ideas borrowed from the staggered grid approach in finite volume. We first apply this extension to a problem having analytical solution and then to the famous lid-driven square cavity problem. We also apply our formulation to the backward-facing step problem to see how the method performs for external flow problems. The results are presented and are compared with established numerical results. This new approach is seen to produce excellent comparison in all the cases.

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A pure-compact scheme for the streamfunction formulation of Navier–Stokes equations

Cited by 70 publications

References 20 publications

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

A consistent splitting scheme for unsteady incompressible viscous flows I. Dirichlet boundary condition and applications

The (9,5) HOC formulation for the transient Navier–Stokes equations in primitive variable

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