  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.
