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)

Micro-plasticity theories and models are suitable to explain and predict mechanical response of devices on length scales where the influence of the carrier of plastic deformation -the dislocations -cannot be neglected or completely averaged out. To consider these effects without resolving each single dislocation a large variety of continuum descriptions has been developed, amongst which the higher-dimensional continuum dislocation dynamics (hdCDD) theory by Hochrainer et al. (Phil. Mag. 87, pp. 1261-1282) takes a different, statistical approach and contains information that are usually only contained in discrete dislocation models. We present a concise formulation of hdCDD in a general single-crystal plasticity context together with a discontinuous Galerkin scheme for the numerical implementation which we evaluate by numerical examples: a thin film under tensile and shear loads. We study the influence of different realistic boundary conditions and demonstrate that dislocation fluxes and their lines' curvature are key features in small-scale plasticity.

Bioconvection due to microorganisms is important area of research, considerably importance for environment and sustainable fuel cell technologies. Buongiorno nanofluid model for Cattaneo-Christov heat and mass flux theory taken into account the Oldroyd-B nanofluid and gyrotactic microorganisms in a rotating system with the effects of Hall current, and Darcy porous medium is scrutinized. The constitutive equations of the problem are transformed into nondimensional equations with the help of similarity transformations. Homotopy analysis method is used to obtain the solution. Graphs and table support the comprehesive representation of the achieved results. Radial velocity is reduced with the increasing values of relaxation time, retardation time and magnetic field parameters while heat transfer is augmented with thermal relaxation time parameter. The nanoparticles concentration is reduced with the increasing values of Schmidt number and the gyrotactic microorganisms concentration is enhanced with the increasing values of Peclet number. A nice agreement is obtained while comparing the present results numerically with the published results. The proposed mathematical model is used in biochemical engineering, meteorology, power and transportation production, optoelectronic and sensing microfabrication.

The reformulation in conservative form of the higher-dimensional continuum dislocations dynamics(hdCDD) theory of Hochrainer (Ph.D. thesis, 2006) is presented together with a framework for elasto-plasticity problem based on this theory. A Runge-Kutta discontinuous Galerkin(RKDG) method is used for the evolution of hdCDD to obtain information of the micro-structure which is coupled with a finite element method for the stress computation.

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