Efficient time integration for discontinuous Galerkin approximations of linear wave equations

Abstract

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro-magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization.For the time integration we discuss advantages and disadvantages of explicit and implicit Runge-Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves.∂ t E(u(t)) = M∂ t u(t), u(t) 0,Ω = − Au(t), u(t) 0,Ω = 0, i.e., the energy is conserved E(u(t)) = E(u(0)) for all t ∈ [0, T ].We study three different applications fitting into this framework, namely acoustic, elastic, and electro-magnetic waves. ApplicationsIn all applications, the operator A corresponds to a linear system of J first-order differential equations.Acoustic waves Acoustic waves in an isotropic medium with variable density ρ ∈ L ∞ (Ω) are described by the secondorder scalar equation for the potentialWe assume ρ(x) ≥ ρ 0 > 0 for a.a. x ∈ Ω. Introducing the pressure p = ∂ t ψ and the flux q = −∇ψ this corresponds to the first-order system ρ∂ t p + div q = 0 , ∂ t q + ∇p = 0 with J = D + 1 components. We define the operators M (q, p) = (q, ρp), A(q, p) = (∇p, div q)

Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling

“…In this case system (2) can be uncoupled by an orthogonal transformation into a set of harmonic oscillators, see [25]. Application of the Krylov/SAI type of methods to electromagnetic problems has recently been discussed in [18,19]. The focus of [18] is on discontinuous Galerkin formulations for various wave problems, where the considered exponential time integration methods employ matrix exponentials within each time step.…”

“…Application of the Krylov/SAI type of methods to electromagnetic problems has recently been discussed in [18,19]. The focus of [18] is on discontinuous Galerkin formulations for various wave problems, where the considered exponential time integration methods employ matrix exponentials within each time step. Our approach discussed here is aimed at using matrix exponential across time steps, which, in our experience, often leads to a better efficiency.…”

Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems

“…is the upwind flux obtained from local solutions of Riemann problems, see [20,Section 2]. Again using integration by parts, we obtain…”

“…For the examples in Section 2, the numerical upwind flux in homogeneous media is given as follows (see [20] for the explicit solution of Riemann problems in heterogeneous media).…”

“…In Section 2 we introduce a setting for linear hyperbolic operators by reference to applications in the field of linear transport and acoustic and electro-magnetic waves, and we establish the well-posedness of the space-time variational problem based on a technique developed in [31]. In Section 3, following the setting established in [20], we consider a semi-discrete discontinuous Galerkin discretization in spatial direction with upwind flux. On this basis we define an implicit Petrov-Galerkin space-time discretization in Section 4, and we prove wellposedness of the discrete method and convergence on tensor product space-time meshes.…”