Eikonal方程的两类高阶快速扫描格式  

Two Classes of High-order Fast Sweeping Schemes for Eikonal Equations

作　　者：黄晓倩 蒋艳群 胡迎港 蒋剑军 HUANG Xiaoqian;JIANG Yanqun;HU Yinggang;JIANG Jianjun(School of Information Engineering,Southwest University of Science and Technology,Mianyang 621010;School of Mathematics and Physics,Southwest University of Science and Technology,Mianyang 621010;School of Mathematics and Computer Science,Tongling University,Tongling 244061)

机构地区：[1]西南科技大学信息工程学院,绵阳621010 [2]西南科技大学数理学院,绵阳621010 [3]铜陵学院数学与计算机学院,铜陵244061

出　　处：《工程数学学报》2025年第2期297-310,共14页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11872323);国家数值风洞工程项目(NNW2018-ZT4A08);安徽省高校自然科学重点研究项目(KJ2020A0973)。

摘　　要：Eikonal方程在计算机视觉、图像处理、几何光学等领域中有着广泛应用。将高阶精度加权紧致非线性格式(Weighted Compact Nonlinear Scheme, WCNS)和加权基本无振荡(Weighted Essentially Non-oscillatory, WENO)格式推广用于求解Eikonal方程,设计了高阶快速扫描WCNS格式和高阶快速扫描WENO格式。将稳态Eikonal方程转化为伪时间相关问题,具有单调性的Lax-Friedrichs型格式用于计算数值哈密顿通量,五阶WCNS格式和五阶WENO格式用于计算未知变量的空间导数的左右极限值。为加快算法收敛速度以及避免求解离散形式的非线性系统,伪时间方向上采用结合了快速扫描策略的显式时间离散格式。数值结果表明,快速扫描WCNS格式和快速扫描WENO格式在光滑区均能达到五阶设计精度,两者得到的数值解与方程精确解吻合很好。此外,两种格式的计算效率比同阶经典WENO格式要高。Eikonal equations are widely used in computer vision,image processing,geomet-ric optics,etc.This paper extends the weighted compact nonlinear scheme(WCNS)and the weighted essentially non-oscillatory(WENO)scheme for hyperbolic conservation laws and de-signs high-order fast sweeping WCNS and WENO schemes to solve the pseudo-time dependent Eikonal equations.Fifth-order WCNS and WENO schemes are applied to compute the left and right limit values of spatial derivatives of the unknown variable coupled with the monotone Lax-Friedrichs numerical Hamiltonians.In order to speed up the convergence of the designed algorithm and to avoid solving a nonlinear discrete system,an explicit time-marching scheme combined with a fast sweeping strategy is used for time discretization.Numerical results show that both the fast sweeping WCNS method and the fast sweeping WENO method can achievefifth-order accuracy in smooth regions and the numerical solutions obtained with the two meth-ods are in good agreement with the exact solutions of Eikonal equations.Compared with the classical WENO method of the same order,the fast sweeping WCNS and WENO schemes are more efficient when they obtain the same numerical errors.

关 键 词：Eikonal方程 WCNS格式 WENO格式 快速扫描方法 

分 类 号：O241.8[理学—计算数学]