Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

Sabetghadam, Fereidoun; Sharafatmandjoor, Shervin; Norouzi, Farhang

doi:10.1016/j.jcp.2008.08.018

Cited by 28 publications

(37 citation statements)

References 17 publications

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“…Here ∂ j ξ/∂n j denotes the j th normal derivative of ξ on the boundary Γ; a computational formula for ∂ j ξ/∂n j is given later in Equation (22). A simple choice of the operator H k is the polyharmonic operator H k = ∆ k+1 .…”

Section: Smooth Extension Of a Known Function From ω To Cmentioning

confidence: 99%

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Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

Stein

Guy

Thomases

2016

Journal of Computational Physics

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The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems.

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Section: Smooth Extension Of a Known Function From ω To Cmentioning

confidence: 99%

“…The advantages of EB methods are substantial enough that significant effort has been expended on improving their accuracy [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Two different approaches are generally taken.…”

Section: Introductionmentioning

confidence: 99%

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

Stein

Guy

Thomases

2016

Journal of Computational Physics

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“…(For an overview, see Refs. [4,5,6,7,8,9].) To impose boundary conditions at the immersed interfaces, a discretized Dirac delta function is employed to distribute a singular source over nearby grid points.…”

Section: Introductionmentioning

confidence: 99%

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

Chen

Thornton

2012

Modelling Simul. Mater. Sci. Eng.

109

113

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In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelled droplet.

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“…Most EB methods are explicit and of first or second-order of spatial accuracy; typically EB formulations of higher-order of accuracy have not proven reliably stable for non-rectangular domains. A Fourier-based EB method presented in [47] for the Poisson equation has demonstrated some accuracy for a slowly-oscillatory sinusoidal solution on a smooth four-leaf shaped geometry. )…”

Section: Introductionmentioning

confidence: 99%

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

Bruno

Lyon

2010

Journal of Computational Physics

114

199

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Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

Cited by 28 publications

References 17 publications

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

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