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.

翻译：我们提出了一种针对非均匀介质中自伴时变波动方程的新型Hadamard积分器。首先，基于Gelfand-Shilov函数构建新的渐近级数（称为Hadamard拟设），用以近似时变波动方程的格林函数。其次，将Hadamard拟设的首项代入基尔霍夫-惠更斯表示，针对时变波动方程的柯西问题发展了原创性Hadamard积分器，并在测地极坐标中推导出相应的拉格朗日公式。第三，为高效构建拉格朗日框架下的Hadamard积分器，采用短时射线追踪方法精确获取波前位置，并进一步开发快速算法计算基于切比雪夫多项式的低秩表示，涵盖波前位置及Hadamard系数变体。第四，借助这些低秩表示，运用Hadamard积分器高效求解具有高度振荡初始条件的时变波动方程，其中时间步长与初始条件无关。通过合理选择介质相关的时间步长，该新型Hadamard积分器能隐式地传播焦散面后的波场，并自然地在时间上推进空间翻转波。此外，由于积分器与初始条件无关，一旦构建完成即可应用于多种不同初始条件。二维和三维数值算例验证了所提方法的精度与性能。