Analytical and Numerical Analysis of Linear and Nonlinear Properties of an rf-SQUID Based Metasurface

We derive a model to describe the interaction of an rf-SQUID (radio frequency superconducting quantum interference device) based metasurface with free space electromagnetic waves. The electromagnetic fields are described on the base of Maxwell's equations. For the rf-SQUID metasurface we rely on an equivalent circuit model. After a detailed derivation, we show that the problem that is described by a system of coupled differential equations is wellposed and, therefore, has a unique solution. In the small amplitude limit, we provide analytical expressions for reflection, transmission, and absorption depending on the frequency. To investigate the nonlinear regime, we numerically solve the system of coupled differential equations using a finite element scheme with transparent boundary conditions and the Crank-Nicolson method. We also provide a rigorous error analysis that shows convergence of the scheme at the expected rates. The simulation results for the adiabatic increase of either the field's amplitude or its frequency show that the metasurface's response in the nonlinear interaction regime exhibits bistable behavior both in transmission and reflection.Comment: published in Physical Review B, Phys. Rev. B 99, 07540

New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation

A propagation method for the time dependent Schr\"odinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde's usually results in system of ode's of the form u_t -Gu = s where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modified exponential evolution operator, had been developed already for this type of problems where G is a linear, time independent matrix and s is a constant vector. In this paper we will describe a new algorithm for the more general case where s is a time-dependent r.h.s vector. An iterative version of the new algorithm can be applied to the general case where G depends on t or u. Numerical results for Schr\"odinger equation with time-dependent potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page

Character of Christ: A Proposal for Excellence in Christian Character Education

Moral teaching programs, such as character education, have been implemented nationwide in order to curb the growing trend of violence, abuse, and moral relativism within schools, both public and private. These programs represent a variety of moral training philosophies, and current research is revealing some best practices within the field. However, these programs do little to address the needs of distinctively Christian educators who seek to train their students toward the character of Jesus Christ. The research in this study promotes the development of a curriculum to meet this need. The following research indicates that character education\u27s premise and many of its practices are worthy of consideration when developing a Christian character curriculum. However, the foundation of the character traits promoted by a Christian character curriculum must not be based on the consensus of a pluralistic society. The foundation must be established solely on the person of Christ. Best practices within the field of character education are emerging through current research. These practices and the theories behind them are also examined in light of the development of a Christian character curriculum. Recommendations and implications for a Christian character curriculum are made in both theory and practice

Preconditioned implicit time integration schemes for Maxwell’s equations on locally refined grids

In this paper, we consider an efficient implementation of higher-order implicit time integration schemes for spatially discretized linear Maxwell’s equations on locally refined meshes. In particular, our interest is in problems where only a few of the mesh elements are small while the majority of the elements is much larger. We suggest to approximate the solution of the linear systems arising in each time step by a preconditioned Krylov subspace method, e.g., the quasi-minimal residual method by Freund and Nachtigal [13]. Motivated by the analysis of locally implicit methods by Hochbruck and Sturm [25], we show how to construct a preconditioner in such a way that the number of iterations required by the Krylov subspace method to achieve a certain accuracy is bounded independently of the diameter of the small mesh elements. We prove this behavior by using Faber polynomials and complex approximation theory. The cost to apply the preconditioner consists of the solution of a small linear system, whose dimension corresponds to the degrees of freedom within the fine part of the mesh (and its next coarse neighbors). If this dimension is small compared to the size of the full mesh, the preconditioner is very efficient. We conclude by verifying our theoretical results with numerical experiments for the fourth-order Gauß-Legendre Runge–Kutta method

The norm convergence of a Magnus expansion method

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method will be investigated.Comment: Rerferee recommendations incorporated. To appear in Central European Journal of Mathematic

Error analysis of discontinuous Galerkin discretizations of a class of linear wave-type problems

In this paper we consider central fluxes discontinuous Galerkin space discretizations of a general class of wave-type equations of Friedrichs’ type. This class includes important examples such as Maxwell’s equations and wave equations. We prove an optimal error bound which holds under suitable regularity assumptions on the solution. Our analysis is performed in a framework of evolution equations on a Hilbert space and thus allows for the combination with various time integration schemes