The defining feature of waves is their ability to propagate over vast distances in space and time without changing shape. This unique property enables the transfer of information and constitutes the foundation of today’s communication based society. To see that accurate propagation of waves requires high order accurate numerical methods, consider the problem of propagating a wave in three dimensions for 100 wavelengths with 1% error. Using a second order method this requires 0.2 trillion space-time samples while a high order method requires many orders of magnitude fewer samples.
In the first part of this talk we present new arbitrary order dissipative and conservative Hermite methods for the scalar wave equation. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve space-time accuracy of order O(2m). Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for CFL = 1, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.
In the second part of the talk we present a spatial discontinuous Galerkin discretization of wave equations in second order form that relies on a new energy based strategy featuring a direct, mesh-independent approach to defining interelement fluxes. Both energy-conserving and upwind discretizations can be devised. The method comes with optimal a priori error estimates in the energy norm for certain fluxes and we present numerical experiments showing that optimal convergence for certain fluxes.