A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

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.     

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.     

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.     

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.     

Computational models for weakly dispersive nonlinear water waves

Numerical methods for the two- and three-dimensional Boussinesq equations governing weakly nonlinear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by finite element discretization in space. Staggered finite difference schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. Efficient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic finite elements, the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Verification of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a significantly increased computational efficiency when using higher-order schemes and bathymetry-adapted finite element grids.     

High-order Compact Schemes for Nonlinear Dispersive Waves

High-order compact finite difference schemes coupled with high-order low-pass filter and the classical fourth-order Runge–Kutta scheme are applied to simulate nonlinear dispersive wave propagation problems described the Korteweg-de Vries (KdV)-like equations, which involve a third derivative term. Several examples such as KdV equation, and KdV-Burgers equation are presented and the solutions obtained are compared with some other numerical methods. Computational results demonstrate that high-order compact schemes work very well for problems involving a third derivative term.     

CABARET scheme in vorticity-velocity variables for the numerical modeling of ideal fluid motion in a two-dimensional domain

A modification of the CABARET scheme is proposed for the numerical solution of equations of ideal fluid motion in vorticity-velocity variables. The dissipative and dispersive properties of the obtained numerical algorithm were investigated for the problem of an isolated vortex. Calculations were performed for decaying homogeneous isotropic turbulence on grids of varying density. In all investigated grids, the spectral density of the kinetic energy was found to obey the ??-3?? law, which conforms to the Kraichnan-Batchelor theory. The structural functions of the obtained vortex flow conform to the law derived using the dimension theory.     

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space     

Control of numerical effects of dispersion and dissipation in numerical schemes for efficient shock-capturing through an optimal Courant number

Composite schemes consist of several steps of a dispersive scheme followed by one step of a dissipative scheme [Liska Richard, Wendroff Burton. Composite schemes for conservation laws. SIAM J Numer Anal 1998;35(6):2250-71]. The latter [Liska Richard, Wendroff Burton. 2D shallow water equations by composite schemes. Int J Numer Meth Fluids 1999;30:461-79] acts as a filter reducing oscillations in regions of discontinuity. Liska and Wendroff have derived the composite Lax-Wendroff/Lax-Friedrichs (LWLF) [Liska Richard, Wendroff Burton, 1998] scheme which blends the Lax-Wendroff (LW) scheme with the 2-step Lax-Friedrichs (LF) scheme. The formulation of the 2-step Lax-Friedrichs scheme [Liska Richard, Wendroff Burton, 1998] is different from that of the classic Lax-Friedrichs scheme and has been devised by Liska [Liska Richard, Wendroff Burton, 1998]. In this work, we propose to replace LW scheme by MacCormack (MC) scheme since the latter is less dispersive. We obtain a new composite scheme in 1-D and in 2-D by blending the MacCormack scheme with the 2-step Lax-Friedrichs scheme which we term as the composite MacCormack/Lax-Friedrichs (MCLF) scheme. This is followed by analytical work on the effective amplification factor (EAF) and the relative phase error (RPE) for both families of schemes in 1-D and 2-D: LWLFn and MCLFn, consisting of (n − 1) steps of the dispersive scheme (LW or MC) and 1 step of the dissipative LF scheme. We introduce a new concept, baptised as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES in which a cfl number is computed whereby dissipation curbs dispersion. This cfl number is termed as optimal in this work. We conduct a comparative study based on numerical experiments in 2-D namely: contact-discontinuity problem [Ould Kaber SM. A legendre pseudospectral viscosity method. J Comput Phys 1996;128:165-80], rotating hill problem [Ould Kaber SM, 1996] and the deformative flow of Smolarkiewicz [Dabdub Donald, Seinfeld John H. Numerical advective schemes used in air quality models-sequential and parallel implementation. Atmos Environ 1994;28(20):3369-85, Ghods A, Sobouti F, Arkani-Hamed J. An improved second order method for solution of pure advection problems. Int J Numer Meth Fluids 2000;32:959-77, Nguyen K, Dabdub D. Two-level time-marching scheme using splines for solving the advection equation. Atmos Environ 2001;35:1627-37] to show that the MacCormack/Lax-Friedrichs (MCLF) scheme is more efficient than LWLF scheme to capture shocks in regions of discontinuity. We also show that better results are obtained at optimal cfl numbers for some variants of LWLFn and MCLFn schemes, with n = 2, 3, 4 and 5.     

Inverse reactive transport simulator (INVERTS): an inverse model for contaminant transport with nonlinear adsorption and source terms

A numerically based simulator was developed to assist in the interpretation of complex laboratory experiments examining transport processes of chemical and biological contaminants subject to nonlinear adsorption and/or source terms. The inversion is performed with any of three nonlinear regression methods, Marquardt–Levenberg, conjugate gradient, or quasi-Newton. The governing equations for the problem are solved by the method of finite-differences including any combination of three boundary conditions: 1) Dirichlet, 2) Neumann, and 3) Cauchy. The dispersive terms in the transport equations were solved using the second-order accurate in time and space Crank–Nicolson scheme, while the advective terms were handled using a third-order in time and space, total variation diminishing (TVD) scheme that damps spurious oscillations around sharp concentration fronts. The numerical algorithms were implemented in the computer code INVERTS, which runs on any standard personal computer. Apart from a comprehensive set of test problems, INVERTS was also used to model the elution of a nonradioactive tracer, ¹⁸⁵Re, in a pressurized unsaturated flow (PUF) experiment with a simulated waste glass for low-activity waste immobilization. Interpretation of the elution profile was best described with a nonlinear kinetic model for adsorption.     

Dispersion free wave splittings for structural elements

Wave splittings are derived for three types of structural elements: membranes, Timoshenko beams, and Mindlin plates. The Timoshenko beam equation and the Mindlin plate equation are inherently dispersive, as is each Fourier component of the membrane equation in an angular decomposition of the field. The distinctive feature of the wave splittings derived in the present paper is that, in homogeneous regions, they transform the dispersive wave equations into simple one-way wave equations without dispersion. Such splittings have uses both for radial scattering problems in the 2D cases and for scattering problems in dispersive media. As an example of how the splittings may be applied, a direct scattering problem is solved for a membrane with radially varying density. The imbedding method is utilized, and agreement is obtained with an FE simulation.     

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.     

A highly accurate alternating 6-point group method for the dispersive equation

In this paper, we construct a group of Saul'yev type asymmetric difference formulas for the dispersive equation. Based on these formulas we derive a new alternating 6-point group algorithm to solve dispersive equations with periodic boundary conditions. The algorithm has a high-order accuracy in space and an unconditional stability. The theoretical results are conformed to the numerical simulation. A comparison of this algorithm with the previous Alternating Group Explicit method is presented.     

Instability in leapfrog and forward–backward schemes: Part II: Numerical simulations of dam break

Following Sun’s approach [17], Shuman smoothing instead of conventional diffusion terms is used in a simple two-time step semi-implicit finite volume scheme to simulate dam break. When the Courant number is less than one, the absolute value of amplification factor of the 1D linearized shallow-water equations is 1 in this new scheme. Compared with the characteristic-based semi-Lagrangian schemes and the Riemann solver, this scheme produces excellent results of free water depth and speed of the shock. Numerical simulations show that the water inside the dam initially moves away radially until water almost depletes near the center; then the water moves back to the center and forms a vertical water column there. This paper proves that Shuman smoothing can be used not only in the linearized shallow-water equations discussed in Sun [17] but also in the nonlinear wave equations to control instability around shocks.     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

Asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry

The asymptotic derivation of a new family of one-dimensional, weakly nonlinear and weakly dispersive equations that model the flow of an ideal fluid in an elastic vessel is presented. Dissipative effects due to the viscous nature of the fluid are also taken into account. The new models validate by asymptotic reasoning other non-dispersive systems of equations that are commonly used, and improve other nonlinear and dispersive mathematical models derived to describe the blood flow in elastic vessels. The new systems are studied analytically in terms of their basic characteristic properties such as the linear dispersion characteristics, symmetries, conservation laws and solitary waves. Unidirectional model equations are also derived and analysed in the case of vessels of constant radius. The capacity of the models to be used in practical problems is being demonstrated by employing a particular system with favourable properties to study the blood flow in a large artery. Two different cases are considered: A vessel with constant radius and a tapered vessel. Significant changes in the flow can be observed in the case of the tapered vessel.     

一类Lagrange坐标系下的ENO有限体积格式

本文首先从积分形式的二维Lagrange流体力学方程组出发,使用ENO高阶插值多项式,推广了四边形结构网格下的一阶有限体积格式,构造得到了一类结构网格下的高精度有限体积格式.该格式针对单介质问题具有良好的计算效果,同时在处理多介质问题时,不会产生物质界面附近强烈的震荡.结合有效的守恒重映方法,用ALE方法进行数值模拟,得到了预期的效果.     

A Discontinuous Spectral Element Model for Boussinesq-Type Equations

We present a discontinuous spectral element model for simulating 1D nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The advective fluxes are calculated using an approximate Riemann solver while the dispersive fluxes are obtained by centred numerical fluxes. Numerical computation of solitary wave propagation is used to prove the exponential convergence.     

A Dual-Petrov-Galerkin Method for the Kawahara-Type Equations

An efficient and accurate numerical scheme is proposed, analyzed and implemented for the Kawahara and modified Kawahara equations which model many physical phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. The scheme consists of dual-Petrov-Galerkin method in space and Crank-Nicholson-leap-frog in time such that at each time step only a sparse banded linear system needs to be solved. Theoretical analysis and numerical results are presented to show that the proposed numerical is extremely accurate and efficient for Kawahara type equations and other fifth-order nonlinear equations. This work is partially supported by the National Science Council of the Republic of China under the grant NSC 94-2115-M-126-004 and 95-2115-M-126-003. This work is partially supported by NSF grant DMS-0610646.