In this paper, we present a one parameter family of fully discrete Weighted Sequential Splitting (WSS)-finite difference time-domain (FDTD) methods for Maxwell’s equations in three dimensions. In one time step, the Maxwell WSS-FDTD schemes consist of two substages each involving the solution of several 1D discrete Maxwell systems. At the end of a time step we take a weighted average of solutions of the substages with a weight parameter $θ$, $0 ≤ θ ≤ 1$. Similar to the Yee-FDTD method, the Maxwell WSS-FDTD schemes stagger the electric and magnetic fields in space in the discrete mesh. However, the Crank-Nicolson method is used for the time discretization of all 1D Maxwell systems in our splitting schemes. We prove that for all values of $θ$, the Maxwell WSS-FDTD schemes are unconditionally stable, and the order of accuracy is of first order in time when $θ\neq 0.5$, and of second order when $θ = 0.5$. The Maxwell WSS-FDTD schemes are of second order accuracy in space for all values of $θ$. We prove the convergence of the Maxwell WSS-FDTD methods for all values of the weight parameter $θ$ and provide error estimates. We also analyze the discrete divergence of solutions to the Maxwell WSS-FDTD schemes for all values of $θ$ and prove that for $θ\neq 0.5$ the discrete divergence of electric and magnetic field solutions is approximated to first order, while for $θ = 0.5$ we obtain a third order approximation to the exact divergence. Numerical experiments and examples are given that illustrate our theoretical results.

In this paper, we analyze the local discontinuous Galerkin method with the generalized alternating numerical flux for two-dimensional singularly perturbed problem with outflow boundary layers. By virtue of the two-dimensional generalized Gauss-Radau projection and energy technique with suitable weight function, we obtain the double-optimal error estimate, namely, the convergence rate in L2-norm out of the outflow boundary layer is optimal, and the width of boundary layer is quasi-optimal, when piecewise tensor product polynomial space on quasi-uniform Cartesian meshes are used. Numerical experiments are given to verify the theoretical results.

This paper reports on a computational study of the model error in the LANS-alpha and NS-alpha deconvolution models of homogeneous isotropic turbulence. Computations are also performed for a new turbulence model obtained as a rescaled limit of the deconvolution model. The technique used is to plug a solution obtained from direct numerical simulation of the incompressible Navier–Stokes equations into the competing turbulence models and to then compute the time evolution of the resulting residual. All computations have been done in two dimensions rather than three for convenience and efficiency. When the effective averaging length scale in any of the models is $α_0 = 0.01$, the time evolution of the root-mean-squared residual error grows as $\sqrt{t}$. This growth rate similar to what would happen if the model error were given by a stochastic force. When $α_0 = 0.20$, the residual error grows linearly. Linear growth suggests that the model error possesses a systematic bias. Finally, for $α_0 = 0.04$, the residual error in LANS-alpha model exhibited linear growth; however, for this value of $α_0$, the higher-order alpha models that were tested did not.

In this paper, a novel second-order two-scale (SOTS) analysis method and corresponding numerical algorithm is developed for dynamic thermo-mechanical problems of composite structures with cylindrical periodicity. The formal SOTS solutions are successfully constructed by the multiscale asymptotic analysis. Then we theoretically explain the necessity of developing the SOTS solutions by the error analysis in the pointwise sense. Furthermore, the convergence result with an explicit rate for the SOTS solutions is obtained. In addition, a SOTS numerical algorithm is presented to effectively solve these multiscale problems. Finally, some numerical examples verify the feasibility and validity of the SOTS numerical algorithm we proposed. This study offers a unified multiscale framework that enables the simulation and analysis of thermo-mechanical coupled behavior of composite structures with cylindrical periodicity.

We address the problem of controlling the unsteady flow separation over an aerofoil, using plasma actuators. Despite the complexity of the dynamics of interest, we show how the problem of controlling flow separation can be formulated as a simple set-point tracking problem, so that a simple control strategy may be used. A robust output feedback control is designed, on the basis of a low-order, linear, dynamical model approximating the incompressible Navier-Stokes equations, obtained from the snapshots of 2D laminar finite element simulations at $Re = 1, 000$. Fast flow reattachment is achieved, along with both stabilisation and increase/reduction of the lift/drag, respectively. Accurate 2D finite element simulations of the full-order nonlinear equations illustrate the effectiveness of the proposed approach: good dynamic performances are obtained, as both the Reynolds number and the angle of attack are varied. The chosen output can be experimentally measured by appropriate sensors and, despite its simplicity, the proposed set-point tracking controller is sufficient to suppress the laminar separation bubble; moreover, its extension to 3D turbulent configurations is straightforward ([33; 7]), thus illustrating the effectiveness of the designed control algorithm in more practical conditions, which are far from the design envelope.

We present a general approach to deriving energy stable numerical approximations for thermodynamically consistent models for nonequilibrium phenomena. The central idea behind the systematic numerical approximation is the energy quadratization (EQ) strategy, where the system's free energy is transformed into a quadratic form by introducing new intermediate variables. By applying the EQ strategy, one can develop linear, high order semi-discrete schemes in time that preserve the energy dissipation property of the original thermodynamically consistent model equations. The EQ method is developed for time discretization primarily. When coupled with an appropriate spatial discretization, a fully discrete, high order, linear scheme can be developed to warrant the energy dissipation property of the fully discrete scheme. A host of examples for phase field models are presented to illustrate the effectiveness of the general strategy.