Single cell discretizations of order two and four for biharmonic problems

Stephenson, J. W.

doi:10.1016/0021-9991(84)90015-9

Cited by 107 publications

(69 citation statements)

References 6 publications

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“…Certain second and fourth-order finite difference approximations for the biharmonic equation (1.1) on a 9-point compact stencil are given in [29]. The approach there involves discretizing the biharmonic equation (1.1) using not only the grid values of the unknown solution u(x, y) but also the values of the gradients u x (x, y) and u y (x, y) at selected grid points.…”

Section: A Brief Review Of Related Finite Difference Methodsmentioning

confidence: 99%

A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

Chen¹,

Li²,

Lin³

2004

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Biharmonic equations have many applications, especially in fluid and solid mechanics, but difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity ∆u along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical example show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.

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Section: A Brief Review Of Related Finite Difference Methodsmentioning

confidence: 99%

A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

Chen¹,

Li²,

Lin³

2004

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“…See [45,33] for a review on fundamental formulations of incompressible Navier-Stokes equations. The appearance and growing popularity of "compact schemes" brought a renewed interest in the aforementioned methods ( [26,17,18,43,42,19,50,35,1,13]). The purestreamfunction formulation for the time-dependent Navier-Stokes system in planar domains has been used in [31,32,30] some twenty years ago.…”

Section: Introductionmentioning

confidence: 99%

“…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. In addition, a fast solver for the fourth order elliptic problems, which is applied at each time step, is presented in [8].…”

Section: Introductionmentioning

confidence: 99%

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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“…The advantage of the streamfunction formulation of the Navier-Stokes equations is that there is no need to invoke inter-functions relations. Our scheme is based on Stephenson's [7] scheme for the biharmonic equation, where the values of the streamfunction ψ and its first-order derivatives ψ x and ψ y serve as interpolated values. We then show that the convective term may be approximated by standard finite difference schemes applied on the first-order derivatives of ψ.…”

Section: Introductionmentioning

confidence: 99%

“…Notice that we adopt, as in [7] the notation ψ x for the difference operator (4.2-c), which should not be confused with the operator ∂ x since it operates on discrete functions. Notice also that we adopt only for convenience the notation δ 4 x , but we do not have δ…”

Section: Introductionmentioning

confidence: 99%

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

Fishelov

Ben‐Artzi

Croisille

2003

Computational Science and Its Applications — ICCSA 2003

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Abstract. We introduce a pure-streamfunction formulation for the incompressible Navier-Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacianof the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function relations need to invoked. A compact numerical scheme, which interpolates streamfunction values as well as its first order derivatives, is presented and analyzed.

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Single cell discretizations of order two and four for biharmonic problems

Cited by 107 publications

References 6 publications

A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

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

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

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