An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge–Kantorovich optimization

Delzanno, Gian Luca; Chacòn, Luis; Finn, John M.; Chung, Yeojin; Lapenta, Giovanni

doi:10.1016/j.jcp.2008.07.020

Cited by 73 publications

(148 citation statements)

References 57 publications

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“…Taking the variation of F2 with respect to x', we find, as in Ref. [1] x' = x -V',\(x') , (6) where V'…”

Section: Review Of the L2 Theorymentioning

confidence: 76%

“…If these densities are, respectively, the local errors in a numerical discretization scheme at times t and t + 6..t, the method equidistributes the errors, and it has been argued that this minimizes the total error [4] . This method is optimal in the sense that it minimizes the L2 norm rJ ]1/2 ~p(x)lx' -xl2dx (1) while preserving the Jacobian constraint '( ' ) (ax' ay' ax' a y ') _ ( )…”

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

“…In Ref. [1] it was shown that the MA equation can readily be solved by Newton Krylov methods, and that the ellipticity property means that multigrid pre conditioning can be effectively used [1 ,6,7] .…”

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

“…In a recent paper [1] , the fundamentals of two-dimensional grid generation and adaptation by Monge-Kantorovich optimization were laid out for the unit square X = [0,1] x [0,1]. (In a later work [2], this theory was extended to more general physical domains and to the unit cube in three dimensions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Delzanno¹,

Finn²

2010

SIAM J. Sci. Comput.

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The Monge-Kantorovich grid generation and adaptation scheme of [Delzanno et al, J. Gomput. Phys. 227, 9841 (2008)] is generalized from a variational principle based on L2 to a variational principle based on Lp. A generalized Monge-Ampere (MA) equation is derived and its properties are discussed . Results for p > 1 are obtained and compared in terms of the quality of the resulting grid. We conclude that for the grid generation application, the formulation based on Lp for p close to unity leads to serious problems associated with the boundary. Results for 1.5 :s p :s 2.5 are quite good, but there is a fairly narrow range around p = 2 where the results are close to optimal with respect to grid distortion. Furthermore, the Newton-Krylov methods used to solve the generalized MA equation perform best for p = 2.

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“…Taking the variation of F2 with respect to x', we find, as in Ref. [1] x' = x -V',\(x') , (6) where V'…”

Section: Review Of the L2 Theorymentioning

confidence: 76%

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

Section: Preprint Submitted To Elsevier Sciencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Delzanno¹,

Finn²

2010

SIAM J. Sci. Comput.

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“…The algorithm has also been applied in three space dimensions [91], to track moving shock waves [87], and in image registration [41]. Delzanno et al [51] however note some situations where the quality of the grids obtained compares unfavourably with a Monge-Kantorovich transformation approach.…”

Section: The Deformation Methodsmentioning

confidence: 99%

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Baines

Hubbard

Jimack

2011

Commun. comput. phys.

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Abstract. This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

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On the diffusion algorithm for density‐equalizing maps with piecewise constant initial data

Avinyó

Solà-Morales

València

2012

Math Methods in App Sciences

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We mathematically analyze the diffusion-based algorithm to produce maps with a given Jacobian, introduced independently\ud by M. T. Gastner and M. E. J. Newman (2004) and ourselves (2003), but in particular cases where the initial density\ud has line or angle discontinuities in the plane. In this situation, the conclusion reinforces the conjecture that the algorithm\ud is always well-posed, in accordance with its extensive numerical use in some areas of applied sciences (cartograms, sensor\ud networks, computational grids, or image registration).Peer ReviewedPostprint (published version

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An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge–Kantorovich optimization

Cited by 73 publications

References 57 publications

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Generalized Monge–Kantorovich Optimization for Grid Generation and Adaptation in $L_{p}$

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

On the diffusion algorithm for density‐equalizing maps with piecewise constant initial data

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