Hybrid high-order methods for the wave equation in time regime (Omar Duran, University of Bergen )

A hybrid high-order method (HHO) was developed for the wave equation and analyzed for fitted [1,2]
and unfitted [3] meshes. The presentation discusses methods for fitting and unfitting meshes at material
interfaces. The second-order formulation in time as well as its reformulation as a first-order system
are discussed. For the time discretization of the second-order formulation, the implicit, second-order
accurate Newmark scheme is used, while the diagonally implicit Runge-Kutta scheme is used. Both
HHO schemes are susceptible to static condensation at every time step. Based on numerical results
obtained from test cases involving analytical solutions, it has been demonstrated that the methods are
capable of delivering optimal convergence rates.

[1] Burman, E., Duran, O., & Ern, A. (2021). Hybrid High-Order Methods for the Acoustic Wave
Equation in the Time Domain. In Communications on Applied Mathematics and Computation (Vol.
4, Issue 2, pp. 597–633). Springer Science and Business Media LLC. doi.org/10.1007/s42967-
021-00131-8
[2] Burman, E., Duran, O., Ern, A., & Steins, M. (2021). Convergence Analysis of Hybrid High-Order
Methods for the Wave Equation. In Journal of Scientific Computing (Vol. 87, Issue 3). Springer
Science and Business Media LLC. doi.org/10.1007/s10915-021-01492-1
[3] Burman, E., Duran, O., & Ern, A. (2022). Unfitted hybrid high-order methods for the wave equation.
In Computer Methods in Applied Mechanics and Engineering (Vol. 389, p. 114366). Elsevier BV.
doi.org/10.1016/j.cma.2021.11436