article

A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions

Authors:

Shahnawaz Ahmed,

Joyce McLaughlin,

Daniel RenziAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 33, Issue 5

Pages 2402 - 2420

https://doi.org/10.1137/10080258X

Published: 01 September 2011 Publication History

Abstract

In this paper, we develop a third order accurate fast marching method for the solution of the eikonal equation in two dimensions. There have been two obstacles to extending the fast marching method to higher orders of accuracy. The first obstacle is that using one-sided difference schemes is unstable for orders of accuracy higher than two. The second obstacle is that the points in the difference stencil are not available when the gradient is closely aligned with the grid. We overcome these obstacles by using a two-dimensional (2D) finite difference approximation to improve stability, and by locally rotating the grid 45 degrees (i.e., using derivatives along the diagonals) to ensure all the points needed in the difference stencil are available. We show that in smooth regions the full difference stencil is used for a suitably small enough grid size and that the difference scheme satisfies the von Neumann stability condition for the linearized eikonal equation. Our method reverts to first order accuracy near caustics without developing oscillations by using a simple switching scheme. The efficiency and high order of the method are demonstrated on a number of 2D test problems.

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Cited By

Makmul J(2020)A Cellular Automaton Model for Pedestrians’ Movements Influenced by Gaseous Hazardous Material SpreadingModelling and Simulation in Engineering10.1155/2020/34021982020Online publication date: 25-Jan-2020
https://dl.acm.org/doi/10.1155/2020/3402198
Oberman ASalvador T(2015)Filtered schemes for Hamilton-Jacobi equationsJournal of Computational Physics10.1016/j.jcp.2014.12.039284:C(367-388)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.12.039

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 33, Issue 5

^† Special Section: 2010 Copper Mountain Conference

2011

972 pages

ISSN:1064-8275

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 September 2011

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Cited By

Makmul J(2020)A Cellular Automaton Model for Pedestrians’ Movements Influenced by Gaseous Hazardous Material SpreadingModelling and Simulation in Engineering10.1155/2020/34021982020Online publication date: 25-Jan-2020
https://dl.acm.org/doi/10.1155/2020/3402198
Oberman ASalvador T(2015)Filtered schemes for Hamilton-Jacobi equationsJournal of Computational Physics10.1016/j.jcp.2014.12.039284:C(367-388)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.12.039

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