Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms

Sethian, James A.; Vladimirsky, Alexander

doi:10.1137/s0036142901392742

Cited by 333 publications

(421 citation statements)

References 26 publications

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“…Among such methods are the fast marching method and the fast sweeping method. The fast marching method [48,43,22,44,45] is based on the Dijkstra's algorithm [18]. The solution is updated by following the causality in a sequential way; i.e., the solution is updated pointwise in the order that the solution is strictly increasing (decreasing); hence two essential ingredients are needed in the algorithm: an upwind difference scheme and a heap-sort algorithm.…”

Section: Introductionmentioning

confidence: 99%

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Shu

Zhang

et al. 2008

Journal of Computational Physics

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Section: Introductionmentioning

confidence: 99%

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Shu

Zhang

et al. 2008

Journal of Computational Physics

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“…A detailed description of the numerical details is unfortunately not possible due to space limitations. The main propositions in literature for anisotropic FMM are the ordered-upwind methods [4], a derivation of these [3], a control-theoretic approach [5] and a recursive correction FMM [6]. Based on numerical tests we find a combination of [3] and [6] to be most suitable for the anisotropies we are faced with.…”

Section: Anisotropic Fast Marching Methodsmentioning

confidence: 99%

Blood Particle Trajectories in Phase-Contrast-MRI as Minimal Paths Computed with Anisotropic Fast Marching

Schwenke

Hennemuth

Fischer

et al. 2011

Bildverarbeitung Für Die Medizin 2011

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Abstract. In this paper the well-established minimal path framework is applied to the problem of computing blood flow trajectories in phase contrast magnetic resonance imaging (PC-MRI). The velocity vectors measured by PC-MRI and an uncertainty measure are combined to a metric tensor and minimal paths are calculated with anisotropic fast marching method (FMM). Exemplary results of the computation of the blood particle trajectories are given. This work contributes a novel application of anisotropic FMM to flow computations.

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“…This procedure creates a small initial band that can be input to the efficient Fast Marching Method [26], which is a Dijkstra-like ordered upwind finite difference scheme for solving the full Eikonal equation outside this initial band. A different Dijkstra-like control theoretic discretization of the Eikonal equation stemming from optimal control was developed in [32], and we refer the reader to [29] for a detailed discussion and extensions of Fast Marching Methods to general front propagation problems.…”

Section: Solving the Eikonal Equationmentioning

confidence: 99%

Analysis and applications of the Voronoi Implicit Interface Method

Saye

Sethian

2012

Journal of Computational Physics

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a b s t r a c tWe analyze a new mathematical and numerical framework, the ''Voronoi Implicit Interface Method'' (''VIIM''), first introduced in Saye and Sethian (2011) [R.I. Saye, J.A. Sethian, The Voronoi Implicit Interface Method for computing multiphase physics, PNAS 108 (49) (2011) 19498-19503] for tracking multiple interacting and evolving regions (''phases'') whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.). From a mathematical point of view, the method provides a theoretical framework for moving interface problems that involve multiple junctions, defining the motion as the formal limit of a sequence of related problems. Discretizing this theoretical framework provides a numerical methodolology which automatically handles multiple junctions, triple points and quadruple points in two dimensions, as well as triple lines, etc. in higher dimensions. Topological changes in the system occur naturally, with no surgery required. In this paper, we present the method in detail, and demonstrate several new extensions of the method to different physical phenomena, including curvature flow with surface energy densities defined on a per-phase basis, as well as multiphase fluid flow in which density, viscosity and surface tension can be defined on a per-phase basis.We test this method in a variety of ways. We perform rigorous analysis and demonstrate convergence in both two and three dimensions for a variety of evolving interface problems, including verification of von Neumann-Mullins' law in two dimensions (and its analog in three dimensions), as well as normal driven flow and curvature flow with and without constraints, demonstrating topological change and the effects of different boundary conditions. We couple the method to a second order projection method solver for incompressible fluid flow, and study the effects of membrane permeability and impermeability, large shearing torsional forces, and the effects of varying density, viscosity and surface tension on a per-phase basis. Finally, we demonstrate convergence in both space and time of a topological change in a multiphase foam.

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Ordered Upwind Methods for Static Hamilton--Jacobi Equations: Theory and Algorithms

Cited by 333 publications

References 26 publications

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

Blood Particle Trajectories in Phase-Contrast-MRI as Minimal Paths Computed with Anisotropic Fast Marching

Analysis and applications of the Voronoi Implicit Interface Method

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