Integral Methods in Science and Engineering - Analytic and Computational Procedures
Integral Methods in Science and Engineering - Analytic and Computational Procedures
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科学与工程中的积分方法 - 分析和计算程序
DOI：
10.1007/978-3-031-34099-4_8
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
Chappell D
中科院分区：
文献类型：
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作者：
Chappell D
射线追踪是一种成熟的模拟高频波传播的方法，其中射线轨迹由常微分方程组的哈密顿系统定义。然后，通过估计给定评估点附近的射线密度，得出波幅的近似值。另一种方法是使用Liouville方程直接根据相空间中的射线密度来表示射线跟踪模型。然后可以使用Frobenius-Perron(F-P)运算符以积分形式表示解，该运算符是沿轨迹传输射线密度的传递运算符。对这类算子进行离散化的经典方法可以追溯到1960年斯坦尼斯瓦夫·乌兰的工作。在某些情况下，通常在具有连续密度和双曲动力学的低维环境中，已经建立了ULAM方法的收敛。在这一章中，我们概述了最近研究三角台球中光线跟踪的Ulam方法的收敛的一些工作，在三角台球中，动力学是抛物线的，流图包含跳跃间断。
Ray-tracing is a well established approach for modelling wave propagation at high frequencies, in which the ray trajectories are defined by a Hamiltonian system of ODEs. An approximation of the wave amplitude is then derived from estimating the density of rays in the neighbourhood of a given evaluation point. An alternative approach is to formulate the ray-tracing model directly in terms of the ray density in phase-space using the Liouville equation. The solutions may then be expressed in integral form using the Frobenius-Perron (F-P) operator, which is a transfer operator transporting the ray density along the trajectories. The classical approach for discretising such operators dates back to 1960 and the work of Stanislaw Ulam. The convergence of the Ulam method has been established in some cases, typically in low dimensional settings with continuous densities and hyperbolic dynamics. In this chapter, we outline some recent work investigating the convergence of the Ulam method for ray tracing in triangular billiards, where the dynamics are parabolic and the flow map contains jump discontinuities.