Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation

We describe implicit and explicit formulations of the hybridizable discontinuous Galerkin method for the acoustic wave equation based on state-of-the-art numerical software and quantify their efficiency for realistic application settings. In the explicit scheme, the trace of the acoustic pressure is computed from the solution on the two elements adjacent to the face at the old time step. Tensor product shape functions for quadrilaterals and hexahedra evaluated with sum factorization are used to ensure low operation counts. For applying the inverse mass matrix of Lagrangian shape functions with full Gaussian quadrature, a new tensorial technique is proposed. As time propagators, diagonally implicit and explicit Runge–Kutta methods are used, respectively. We find that the computing time per time step is 25 to 200 times lower for the explicit scheme, with an increasing gap in three spatial dimensions and for higher element degrees. Our experiments on realistic 3D wave propagation with variable material parameters in a photoacoustic imaging setting show an improvement of two orders of magnitude in terms of time to solution, despite stability restrictions on the time step of the explicit scheme. Operation counts and a performance model to predict performance on other computer systems accompany our results.