Hermite Methods for the Scalar Wave Equation

Abstract

Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are presented. Both methods use (m + 1) d degrees of freedom per node for the displacement in d-dimensions; the dissipative and conservative methods achieve orders of accuracy (2m − 1) and 2m, respectively. Stability and error analyses as well as implementation strategies for accelerators are also given.1. Introduction. We construct, analyze, and test arbitrary order dissipative and conservative Hermite methods for the scalar wave equation in a medium with constant speed of sound c. The degrees-of-freedom for Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The dissipative method achieves space-time accuracy of order 2m − 1, while the conservative method has space-time order 2m. Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for c∆t ≤ h, 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.Our primary interest in these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type, where boundary conditions are simple to apply. In previous work [7] we considered this type of hybridization of the Hermite methods for first order system proposed in [11] with nodal discontinuous Galerkin (dG) methods. For wave equations in second order form we envision a similar hybridization where the geometry is handled by, for example, our recently developed discontinuous Galerkin methods for wave equations in second order form [1]. Our dG method has the property that, based on the choice of numerical flux, it is either dissipative or conservative.We provide optimal stability and convergence results for both the conservative and dissipative method for one dimensional periodic domains. The analysis for the dissipative method follows the analysis for first order systems [11] but here it is based on the energy of the wave equation v 2 + |∇u| 2 dx. A difference compared to [11] is that we require that the (polynomial) approximation spaces of the velocity, v, and displacement, u to differ by one degree. Its extension to higher space dimensions, however, does not follow in a straightforward way from the Hermite method for first-order systems, requiring a specialized interpolation scheme to achieve order-independent stability at CFL one.The analysis of the conservative method is new and quite different from that of [11]. Additionally, the analysis is done by introducing what we denote conserved variables, this simplifies the analysis conside...

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