用于含时波动方程的Hadamard积分器：基于射线追踪的拉格朗日表述
Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing
AI总结：
本研究针对非均匀介质中的自伴含时波动方程，提出结合Hadamard假设、拉格朗日表述与短时射线追踪的新型Hadamard积分器，通过低秩表示实现与初始条件无关的高效求解，可处理焦散面与高振荡问题，经二维、三维算例验证有效。
摘要：
我们针对非均匀介质中的自伴含时波动方程，提出了一种新型Hadamard积分器。首先，我们基于Gelfand-Shilov函数构建了一个新的渐近级数，将其命名为Hadamard假设（Hadamard's ansatz），用于逼近含时波动方程的格林函数。其次，我们将Hadamard假设的首项代入Kirchhoff-Huygens表示式，为含时波动方程的柯西问题开发了原创的Hadamard积分器，并推导了其在测地极坐标下的对应拉格朗日表述。第三，为了高效构建拉格朗日表述下的Hadamard积分器，我们采用短时射线追踪方法精确获取波前位置，还进一步开发了快速算法，用于计算波前位置和各类Hadamard系数的基于Chebyshev多项式的低秩表示。第四，借助这些低秩表示，我们应用Hadamard积分器高效求解具有高振荡初始条件的含时波动方程，其时间步长与初始条件无关。通过合理选择依赖于介质的时间步长，我们的新型Hadamard积分器可以隐式地在焦散面之外传播波场，并且能够自然地在时间上推进空间翻转的波。此外，由于该积分器与初始条件无关，一旦构建完成，Hadamard积分器便可应用于多种不同的初始条件。二维和三维数值算例均验证了所提方法的准确性和性能。
英文摘要：
We propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. First, we create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the time-dependent wave equation. Second, incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Third, to construct the Hadamard integrator in the Lagrangian formulation efficiently, we use a short-time ray tracing method to obtain wavefront locations accurately, and we further develop fast algorithms to compute Chebyshev-polynomial based low-rank representations of both wavefront locations and variants of Hadamard coefficients. Fourth, equipped with these low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions, where the time step size is independent of the initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally. Moreover, since the integrator is independent of initial conditions, the Hadamard integrator can be applied to many different initial conditions once it is constructed. Both two-dimensional and three-dimensional numerical examples illustrate the accuracy and performance of the proposed method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
