A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions
A Third Order Accurate Fast Marching Method for the Eikonal Equation in Two Dimensions
复制标题
二维方程方程的三阶精确快速推进方法
DOI：
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发表时间：
2011
影响因子：
3.1
通讯作者：
Daniel Renzi
中科院分区：
文献类型：
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作者：
Shahnawaz Ahmed;Stanley Bak;J. McLaughlin;Daniel Renzi
本文提出了一种求解二维程函方程的三阶精度快速推进法。将快速行进法扩展到更高精度的阶数有两个障碍。第一个障碍是，使用单侧差分格式是不稳定的精度高于二阶。第二个障碍是，当梯度与网格紧密对齐时，差异模板中的点不可用。我们通过使用二维（2D）有限差分近似来提高稳定性，并通过局部旋转网格45度（即，使用沿着对角线的导数）以确保差模板中所需的所有点都可用。我们表明，在光滑区域的全差分模板是用于一个适当的足够小的网格尺寸和差分格式满足冯诺依曼稳定性条件的线性程函方程。我们的方法恢复到焦散线附近的一阶精度，而不产生振荡，通过使用一个简单的切换方案。一些二维测试问题的效率和高阶的方法证明。
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.