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)
In this paper we consider implicit Runge-Kutta methods for the time-integration of linear Maxwell's equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauß and Radau collocation methods. The results for the abstract evolution equation are then combined with a discontinuous Galerkin discretization in space using upwind fluxes. For the case that permeability and permittivity are piecewise constant functions, we show error bounds for the full discretization, where the constants do not deteriorate if the spatial mesh width tends to zero.
We establish error bounds of implicit Runge-Kutta methods for a class of quasilinear hyperbolic evolution equations including certain Maxwell and wave equations. Our assumptions cover algebraically stable and coercive schemes such as Gauß and Radau collocation methods. We work in a refinement of the analytical setting of Kato's well-posedness theory.
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