Omar Duran: Thursday 11 March at 15:00
There are two main approaches to derive a fully discrete method for solving the second-order acoustic wave equation in the time regime. One is to discretize directly the second-order time derivative and the Laplacian operator in space. The other approach is to transform the second-order equation into a first-order hyperbolic system. Firstly, we consider the time second-order form. We devise, analyze the energy-conservation properties, and evaluate numerically a hybrid high-order (HHO) scheme for the space discretization combined with a Newmark-like time-marching scheme. The HHO method uses as discrete unknowns cell- and face-based polynomials of some order 0 ≤ k, yielding for steady problems optimal convergence of order (k + 1) in the energy norm [1]. Secondly, inspired by ideas presented in [2] for hybridizable discontinuous Galerkin (HDG) method and the link between HDG and HHO methods in the steady case [3], first-order explicit or implicit time-marching schemes combined with the HHO method for space discretization are considered. We discuss the selection of the stabilization term and energy conservation and present numerical examples. Extension to the unfitted meshes is contemplated for the acoustic wave equation. We observe that the unfitted approach combined with local cell agglomeration leads to a comparable CFL condition as when using fitted meshes [4].
[1] D.A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics. 14 (2014) 461-472.
[2] M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Computer Methods in Applied Mechanics and Engineering, 300:748–769, March 2016.
[3] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modelling and Numerical Analysis, 50(3):635–650, May 2016.
[4] E. Burman, O. Duran, and A. Ern. Unfitted hybrid high-order methods for the wave equation. Available at https://hal.archives-ouvertes.fr/hal-03086432, 2020.
