Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations
Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations
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点源亥姆霍兹方程的 Babich 展开式和高阶欧拉渐进
DOI：
10.1007/s10915-015-0111-7
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发表时间：
2016
影响因子：
2.5
通讯作者：
R. Burridge
中科院分区：
文献类型：
--
作者：
J. Qian;L. Yuan;Yuan Liu;S. Luo;R. Burridge
非均匀介质中点源的亥姆霍兹方程的解的通常的几何光学展开产生两个方程：走时函数的程函方程和振幅函数的输运方程。然而，两个困难立即出现：一个是如何初始化的振幅在点源的波场是奇异的，另一个是，在偶数维空间中，通常的几何光学展开不产生一个一致的渐近近似接近源。Babich（USSR Comput Math Math Phys 5（5）：247-251，1965）开发了一种基于Hankel的渐近展开，可以轻松克服这两个困难。本文从Babich展开式出发，研究了非均匀介质中Helmholtz方程的高阶欧拉渐近性。程函方程和输运方程均采用高阶Lax-Friedrichs加权无振荡（韦诺）格式求解。我们还证明了程函方程的五阶Lax-Friedrichs韦诺格式在程函光滑时是收敛的。数值例子表明，新的欧拉高阶渐近方法是一致准确的邻域内的源和远离它。
The usual geometrical-optics expansion of the solution for the Helmholtz equation of a point source in an inhomogeneous medium yields two equations: an eikonal equation for the traveltime function, and a transport equation for the amplitude function. However, two difficulties arise immediately: one is how to initialize the amplitude at the point source as the wavefield is singular there; the other is that in even-dimension spaces the usual geometrical-optics expansion does not yield a uniform asymptotic approximation close to the source. Babich (USSR Comput Math Math Phys 5(5):247–251, 1965) developed a Hankel-based asymptotic expansion which can overcome these two difficulties with ease. Starting from Babich’s expansion, we develop high-order Eulerian asymptotics for Helmholtz equations in inhomogeneous media. Both the eikonal and transport equations are solved by high-order Lax–Friedrichs weighted non-oscillatory (WENO) schemes. We also prove that fifth-order Lax–Friedrichs WENO schemes for eikonal equations are convergent when the eikonal is smooth. Numerical examples demonstrate that new Eulerian high-order asymptotic methods are uniformly accurate in the neighborhood of the source and away from it.