Efficient Hadamard Integrators for Time-Dependent Wave Equations in Bounded Domains: Forward and Backward Propagation

Abstract. Motivated by Hadamard integrators that were originally designed for pure initial-value problems of wave equations, we develop novel Hadamard integrators for initial-boundary value problems of wave equations in bounded domains in the high-frequency regime, where inhomogeneous Dirichlet boundary conditions may be imposed. To capture high-frequency wave motion near the boundary, we need to deal with two time-boundary integrals in the Huygens–Kirchhoff representation of the wave solution: the first involves the Green’s function itself, and the second involves its normal derivative. To eliminate the first time-boundary integral, we construct Hadamard ansatz–based mirror Green’s functions that locally satisfy the homogeneous Dirichlet boundary condition so that this integral vanishes. To treat the second time-boundary integral, we use the properties of the Green’s function to derive some computable time-boundary singular integrals. To handle these singular integrals numerically, we develop tensorized Gauss–Legendre quadrature formulas to evaluate these integrals to high-order accuracy. Based on these novel treatments of time-boundary integrals, we employ the ray tracing–based Lagrangian Hadamard integrator to update the wavefield in the rest of the computational domain away from the boundary. Consequently, the proposed method can accurately propagate high-frequency waves in bounded domains, either forward in time or backward in time, on relatively coarse grids that scale linearly with respect to the wavenumber, where caustics are implicitly handled with ease. Numerical examples demonstrate the accuracy and efficiency of the proposed method.