Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes

Abarbanel, Saul; Ditkowski, Adi

doi:10.1006/jcph.1997.5653

Cited by 45 publications

(54 citation statements)

References 3 publications

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“…Since, unlike the cases discussed in [1,2], Eq. (2.1) has a second time derivative, attempts to apply naively the methods presented there fail.…”

Section: Theoretical Framework Of the Methodsmentioning

confidence: 96%

“…In [1][2][3], it was shown how the case of a one-dimensional PDE can be used as a building block for the multidimensional case for constructing error-bounded algorithms over complex geometries with Dirichlet boundary condition. We therefore start with the following one-dimensional problem:…”

Section: Theoretical Framework Of the Methodsmentioning

confidence: 99%

“…In this paper, we consider the application of the method of boundary penalty terms (SAT, see [1][2][3]) to the numerical solution of the wave equation in a finite domain with Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Bounded Error Schemes for the Wave Equation on Complex Domains

2006

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This paper considers the application of the method of boundary penalty terms (SAT) to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g., the staggered Yee scheme)-we achieve a decrease of two orders of magnitude in the level of the L 2 -error.

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“…Since, unlike the cases discussed in [1,2], Eq. (2.1) has a second time derivative, attempts to apply naively the methods presented there fail.…”

Section: Theoretical Framework Of the Methodsmentioning

confidence: 96%

Section: Theoretical Framework Of the Methodsmentioning

confidence: 99%

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Bounded Error Schemes for the Wave Equation on Complex Domains

2006

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“…In this work we present a method based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski [24] [25], for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, that arise when dealing with time dependent domains, including treatment of mergers, and breakups of the moving boundary, which may occur during the course of the simulation.…”

Section: Introductionmentioning

confidence: 99%

“…Ditkowski [24] [25], which generalized [23]. The method has been applied successfully to IBVPs on constant domains by embedding the boundary operator into the numerical scheme in a penalty-based approach, see for example [26], [27], [28], [29].…”

Section: A Brief Introduction To Embedded Finite-differencementioning

confidence: 99%

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

Ditkowski

Harness

2009

J Sci Comput

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This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski [24] [25], for initial boundary value problems on constant irregular domains.We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the stefan problem.Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher then 2 nd -order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies that the numerical solution remains consistent and valid for a long integration time.A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2 nd and a 4 th -order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method.

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A Penalty Method for the Vorticity–Velocity Formulation

Trujillo

Karniadakis

1999

Journal of Computational Physics

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Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes

Cited by 45 publications

References 3 publications

Bounded Error Schemes for the Wave Equation on Complex Domains

Bounded Error Schemes for the Wave Equation on Complex Domains

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

A Penalty Method for the Vorticity–Velocity Formulation

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