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：针对非均匀介质中的自伴时变波动方程，本文提出一种新型Hadamard积分器。首先，我们基于Gelfand-Shilov函数创建了名为Hadamard渐近展开的新型级数，用以逼近时变波动方程的格林函数。其次，将Hadamard渐近展开的首项融入基尔霍夫-惠更斯表示，我们发展了时变波动方程柯西问题的原创Hadamard积分器，并在测地极坐标中导出其对应的拉格朗日公式。第三，为高效构建拉格朗日公式中的Hadamard积分器，我们采用短时射线追踪方法精确获取波前位置，并进一步开发快速算法计算基于切比雪夫多项式的波前位置与Hadamard系数变体的低秩表示。第四，借助这些低秩表示，我们将Hadamard积分器应用于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。通过合理选择依赖介质的时问步长，新型Hadamard积分器可隐式传播越过焦散面的波场，并自然实现空间翻转波的时间推进。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。二维与三维数值算例验证了所提方法的精度与性能。