Finite-difference schemes for anisotropic diffusion

Es, Bram van; Koren, Barry; Blank, H. J. de

doi:10.1016/j.jcp.2014.04.046

Cited by 46 publications

(53 citation statements)

References 54 publications

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“…Second, we consider a simple steady diffusion problem given in Ref. [12] with the following exact solution, which simulates a temperature peak, T exact (x, y) = xy(sin(πx)sin(πy)) ω , x, y ∈ [0, 1],…”

Section: Constant Angle Of Misalignmentmentioning

confidence: 99%

First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

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In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs. [J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

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Section: Constant Angle Of Misalignmentmentioning

confidence: 99%

First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

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“…Anisotropic diffusion is encountered in many fields such as heat conduction in magnetized plasma [8], flows in porous media [2], image processing or oceanic flows [9,22]. We are particularly interested in the anisotropic diffusion in magnetized plasmas.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical simulations for anisotropic diffusion problems have been addressed by a lot of researchers and engineers; see the review in [12]. Methods used today include finite volume method [20,21,24], finite difference method [8], mimetic finite difference method [14], discontinuous Galerkin method [1,7], finite element method [10,13,15,19] and so on. These methods are usually efficient for a selected range of ǫ but loss convergence when ǫ ≪ h (h is the mesh size).…”

Section: Introductionmentioning

confidence: 99%

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Uniformly Convergent Scheme for Strongly Anisotropic Diffusion Equations with Closed Field Lines

Wang¹,

Ying²,

Tang³

2018

SIAM J. Sci. Comput.

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In magnetized plasma, the magnetic field confines particles around field lines. The ratio between the intensity of the parallel and perpendicular viscosity or heat conduction may reach the order of 10 12 . When the magnetic fields have closed field lines and form a "magnetic island", the convergence order of most known schemes depends on the anisotropy strength. In this paper, by integration of the original differential equation along each closed field line, we introduce a simple but very efficient asymptotic preserving reformulation, which yields uniform convergence with respect to the anisotropy strength. Only slight modification to the original code is required and neither change of coordinates nor mesh adaptation is needed. Numerical examples demonstrating the performance of the new scheme are presented.

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“…The equation (1) belongs to a large class of diffusion models with the strongly anisotropic diffusion coefficients from many applications, e.g., image processing [13], flows in porous media [1,8], semiconductor modeling [9], heat conduction in fusion plasmas [12], and so on. Note that here we use ε 2 rather than ε in the diffusion coefficients for the reason we will mention later.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansion with boundary layer analysis for strongly anisotropic elliptic equations

Zhou

2018

Communications in Mathematical Sciences

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In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet boundary condition is conducted for each order in the expansion for the fluctuation part. Our results suggest that the leading order is the combination of the mean part and the composite approximation of the fluctuation part for the general Dirichlet boundary condition.

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Finite-difference schemes for anisotropic diffusion

Cited by 46 publications

References 54 publications

First order hyperbolic approach for Anisotropic Diffusion equation

First order hyperbolic approach for Anisotropic Diffusion equation

Uniformly Convergent Scheme for Strongly Anisotropic Diffusion Equations with Closed Field Lines

Asymptotic expansion with boundary layer analysis for strongly anisotropic elliptic equations

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