Electromagnetic waves in media with permittivity dispersion

A technique for the numerical simulation of electromagnetic wave propagation in materials with permittivity depending on the frequency is presented. The technique is based on numerical solutions of Maxwell equations with additional integral components in the bias current density. The technique to calculate the bias current density in dispersive media is represented and the corresponding modification of finite-difference scheme for Maxwell equations developed earlier is carried out. The electromagnetic pulse propagation in solid-fuel power systems is calculated.     

Unified matrix-exponential FDTD formulations for modeling electrically and magnetically dispersive materials

Unified matrix-exponential finite difference time domain (ME-FDTD) formulations are presented for modeling linear multi-term electrically and magnetically dispersive materials. In the proposed formulations, Maxwell?s curl equations and the related dispersive constitutive relations are cast into a set of first-order differential matrix system and the field?s update equations can be extracted directly from the matrix-exponential approximation. The formulations have the advantage of simplicity as it allows modeling different linear dispersive materials in a systematic manner and also can be easily incorporated with the perfectly matched layer (PML) absorbing boundary conditions (ABCs) to model open region problems. Apart from its simplicity, it has been shown that the proposed formulations necessitate less storage requirements as compared with the well-know auxiliary differential equation FDTD (ADE-FDTD) scheme while maintaining the same accuracy performance.     

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.     

An efficient spectral code for incompressible flows in cylindrical geometries

An efficient and accurate numerical scheme is proposed to solve the incompressible Navier-Stokes equations in a bounded cylinder. The scheme is based on a projection method formulated in primitive variables to maintain the incompressibility constraint, with a second-order semi-implicit scheme for the time integration, and a pseudospectral approximation for the space variables. The Chebyshev-collocation method applied in the radial and axial directions, and the Fourier-Galerkin approximation used in the azimuthal direction lead to a sequence of two-dimensional Helmholtz and Poisson equations for every azimuthal coefficient that are solved by a diagonalization technique. Radial expansions are considered in the diameter of the cell in order to avoid clustering about the axis, and the number of points are selected to ensure that r=0 is not a collocation point. A minimal number of regularity conditions are imposed implicitly at the origin by forcing the proper parity of the Fourier expansions in the radial direction. The method has been tested on analytical solutions and compared with other reliable three-dimensional results. The improvements introduced in the treatment of the spatial discretization reduce significantly the difficulty of implementation of the code, and facilitate the use of high resolutions. Different boundary conditions can also be easily implemented.     

A localized approach for the method of approximate particular solutions

The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems.In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.     

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.     

On the Galerkin/Finite-Element Method for the Serre Equations

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge–Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose very restrictive conditions on the temporal stepsize. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the ‘classical’ Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude.     

Solution of time-domain Maxwell equation with PML by using modified Laguerre polynomials

In this work, a stable numerical algorithm proposed by Chung et al. for the time-domain Maxwell equations is generalized. The time-domain Maxwell equations are solved by expressing the transient behaviors in terms of the modified Laguerre polynomials, and then the original equations of the initial value and boundary value can be transformed into a series of problems independent of the time variable. In this case the method of finite difference （FD）, the finite element method （FEM）, the method of moment （MoM）, etc. or the combination of these methods can be used to solve the problems. Finally, a numerical model is provided for the scattering problem with perfect matched layer （PML） by using FD. The comparison between the results of the proposed method and FDTD is presented to verify the proposed new method.     

Numerical analysis of the micro-Couette flow using a non-Newton–Fourier model with enhanced wall boundary conditions

Non-equilibrium effects exist extensively in microfluidic flows, and the accurate simulation of the Knudsen layer behind them is rather challenging for the linear Newton–Fourier model. In this paper, a high-order reduced model (nonlinear coupled constitutive relations) from Eu’s generalized hydrodynamic equations is applied for the investigation of the micro-Couette flows of diatomic nitrogen and monatomic argon as well as Maxwell and hard-sphere molecules using the MacCormack scheme. In order to simulate the confined flows accurately, a set of enhanced wall boundary conditions based on this model are derived with respect to the degree of non-equilibrium. Both the 1st-order Maxwell–Smoluchowski model and the Langmuir slip model are also investigated. For a large range of Knudsen numbers, the results show that the enhanced boundary conditions make a significant improvement in the prediction of flow profiles, especially the temperature profile. The reason behind that is analyzed in detail. The numerical predictions obtained from the high-order model in conjunction with the enhanced boundary conditions are also compared with DSMC, regularized 13 moment equations, Burnett-type equations as well as Navier–Stokes solutions, which highlight its excellent capability in describing the underlying mechanism of the Knudsen layer in the Couette flow.     

Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme

This paper proposes a split cosine scheme for simulating solitary solutions of the sine-Gordon equation in two dimensions, as it arises, for instance, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation allows for soliton-type solutions, a ubiquitous phenomenon in a large variety of physical problems. The semidiscretization approach first leads to a system of second-order nonlinear ordinary differential equations. The system is then approximated by a nonlinear recurrence relation which involves a cosine function. The numerical solution of the system is obtained via a further application of a sequential splitting in a linearly implicit manner that avoids solving the nonlinear scheme at each time step and allows an efficient implementation of the simulation in a locally one-dimensional fashion. The new method has potential applications in further multi-dimensional nonlinear wave simulations. Rigorous analysis is given for the numerical stability. Numerical demonstrations for colliding circular solitons are given.     

Numerical simulations based on shifted second-order difference/finite element algorithms for the time fractional Maxwell’s system

In this article, mixed element algorithms with second-order time convergence results for the two-dimensional time fractional Maxwell’s equations in the Cole–Cole dispersive medium are developed. Fully discrete mixed element systems with shifted parameters \(\theta\) at time \(t=t_{n-\theta }\), which are constructed by combining the generalized BDF2-\(\theta\) schemes in temporal direction and a mixed element method in space direction, are formulated. For the two-dimensional case of the fractional Maxwell’s system, the algorithm implementation process based on the rectangular subdivision is shown in detail. Finally, two numerical examples are provided to confirm the validity of our method and to analyze the influence of parameters \(\alpha\), \(\theta\) for numerical solutions.

     

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form Δt∝²(Δx) for explicit calculations. A new semi-implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It is based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.     

A Legendre Pseudospectral Penalty Scheme for Solving Time-Domain Maxwell’s Equations

In this paper we present a Legendre pseudospectral algorithm based on a tensor product formulation for solving the time-domain Maxwell equations. Our approach starts by conducting an analysis for finding well-posed boundary operators for the Maxwell equations. We then discuss equivalent characteristic boundary conditions for common physical boundary constraints. These theoretical results are then employed to construct a pseudospectral penalty scheme which is asymptotically stable at the semidiscrete level. Numerical computations based on the proposed scheme are also provided for different cases where exact solutions exist. By measuring the differences between the computed and exact solutions, we observe the expected convergence patterns of the scheme. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

Intelligent computing approach to solve the nonlinear Van der Pol system for heartbeat model

In this work, an intelligent computing algorithm is developed for finding the approximate solution of heart model based on nonlinear Van der Pol (VdP)-type second-order ordinary differential equations (ODEs) using feed-forward artificial neural networks (FF-ANNs) optimized with genetic algorithms (GAs) hybrid through interior-point algorithm (IPA). The mathematical modeling of the system is constructed using FF-ANN models by defining an unsupervised error and unknown weights; the networks are tuned globally with GAs, and local refinement of the results is made with IPA. Design scheme is applied to study the VdP heart dynamics model by varying the pulse shape modification factor, damping coefficients and external forcing factor while keeping the fixed value of the ventricular contraction period. The results of the proposed algorithm are compared with reference numerical solutions of Adams method to establish its correctness. Multiple independent runs are performed for the scheme, and results of statistical analyses in terms of mean absolute deviation, root-mean-square error and Nash–Sutcliffe efficiency illustrate its applicability, effectiveness and reliability.

     

Exact boundary controllability of the second-order Maxwell system: Theory and numerical simulation

The exact controllability of the second order time-dependent Maxwell equations for the electric field is addressed through the Hilbert Uniqueness Method. A two-grid preconditioned conjugate gradient algorithm is employed to inverse the H.U.M. operator and to construct the numerical control. The underlying initial value problems are discretized by Lagrange finite elements and an implicit Newmark scheme. Two-dimensional numerical experiments illustrate the performance of the method.     

Application of spectral forcing in lattice-Boltzmann simulations of homogeneous turbulence

An efficient numerical method for the direct simulation of homogeneous turbulent flow has been obtained by combining a spectral forcing algorithm for homogeneous turbulence with a lattice-Boltzmann scheme for solution of the continuity and Navier–Stokes equations. The spectral forcing scheme of Alvelius [Alvelius K. Random forcing of three-dimensional homogeneous turbulence. Phys Fluids 1999;11(7):1880–89] is used which allows control of the power input by eliminating the force–velocity correlation in the Fourier domain and enables anisotropic forcing. A priori chosen properties such as the Kolmogorov length scale, the integral length scale and the integral time scale are recovered. This demonstrates that the scheme works accurately with the lattice-Boltzmann method and that all specific features of the forcing scheme are recovered in the lattice-Boltzmann implementation.     

A second-order analytic solution for oscillatory wind-induced flow in an idealized shallow lake

We present a second-order analytic solution to the nonlinear depth-integrated shallow water equations for free-surface oscillatory wind-driven flow in an idealized lake. Expressing the solution as an asymptotic expansion in the dimensionless wave amplitude (ζ/h), which is considered to be a small parameter, enables simplification of the governing equations and permits the use of a perturbation approach to solve them.This analytic solution provides a benchmark for testing numerical models. In particular, the main merit of this solution is that it accounts for advective effects, which are typically omitted from analytic solutions of two-dimensional free surface flow. In order to retain these effects in an analytic solution, we restrict our attention to forcing from a monochromatic wind stress, consider a constant depth rectangular lake, and simplify the governing equations by omitting the Coriolis and eddy viscosity terms and using a linearised friction factor. As such, the analytic solution is of limited use for considering real world problems. Due to the complexity of the analytic solution computer code for this solution is available online.Our solution is valid for cases where changes in the water surface level are small compared with the depth of the lake, and the advective terms in the momentum equations are small compared with acceleration terms. We examine the validity of these assumptions for three test cases, and compare the second-order analytic solution to numerical results to verify an existing hydrodynamic model.     

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.