A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

Chen, Guo; Li, Zhilin; Lin, Ping

doi:10.21236/ada444064

Cited by 10 publications

(12 citation statements)

References 23 publications

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“…Note that the augmented approaches have been developed for elliptic interface problems with piecewise constant coefficient [6,14], and the fast algorithms for Poisson and biharmonic equations on irregular domains [2,7,17,16]. However, the augmented approach proposed here is for a more difficult problem and it may be the only to get a second order sharp interface method to solve the Stokes flow with discontinuous viscosity.…”

Section: Once the Augmented Variables ([µU] And [µV]mentioning

confidence: 99%

An Augmented Approach for Stokes Equations With Discontinuous Viscosity and Singular Forces

Li¹,

Ito²,

Lai³

2004

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For Stokes equations with discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, the pressure is known to be discontinuous and the velocity is known to be non-smooth. It has been shown that these discontinuities are coupled together which makes it difficult to obtain accurate numerical solutions. In this paper, a second order accurate numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables which are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that fast Poisson solvers can be used in each iteration. Numerical examples against exact solutions show that the new method has average second order accuracy in the infinity norm, and the number of GMRES iterations is independent of mesh sizes. An example of a moving interface problem is also presented.

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Section: Once the Augmented Variables ([µU] And [µV]mentioning

confidence: 99%

An Augmented Approach for Stokes Equations With Discontinuous Viscosity and Singular Forces

Li¹,

Ito²,

Lai³

2004

Self Cite

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“…The last few decades have seen several approaches for numerically solving the interface equations [1][2][3][4][5][6][7]. Most of the earlier numerical works on such problems involved mainly the use of immersed boundary method or immersed interface method (IIM) on uniform grids, and their global order of accuracy was at most second.…”

Section: Introductionmentioning

confidence: 99%

A class of finite difference schemes for interface problems with an HOC approach

Mittal

Kalita

Ray

2016

Numerical Methods in Fluids

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Summary In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently developed higher‐order compact methodology with special interface treatment for the points just next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first formulate the scheme for one‐dimensional (1D) problems, and then extend it directly to two‐dimensional (2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for both the cases. Finally, we show a new direction of implementing the methodology to 2D problems in cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D and 2D cases, including the flow past circular cylinder governed by the incompressible Navier–Stokes equations. We compare our results with existing numerical and experimental results. In all the cases, our formulation is found to produce better results on coarser grids. For the circular cylinder problem, the scheme used is seen to capture all the flow characteristics including the famous von Kármán vortex street. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Their technique is based on embedding the domain in a Cartesian grid and using a polynomial of sixth degree as an interpolation function. Chen et al [9] suggested a fast Poisson solver using FDM for an irregular domain. Either 13-point or 25-point direct approximation of the biharmonic operator, that could be modified at grid point near the boundary, was proposed.…”

Section: Introductionmentioning

confidence: 99%

Solving biharmonic equation using the localized method of approximate particular solutions

Amazzar²,

Naji³

et al. 2014

International Journal of Computer Mathematics

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Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.

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A Fast Finite Difference Method for Biharmonic Equations on Irregular Domains

Cited by 10 publications

References 23 publications

An Augmented Approach for Stokes Equations With Discontinuous Viscosity and Singular Forces

An Augmented Approach for Stokes Equations With Discontinuous Viscosity and Singular Forces

A class of finite difference schemes for interface problems with an HOC approach

Solving biharmonic equation using the localized method of approximate particular solutions

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