Numerical Dispersion Analysis Research Articles

Numerical Stability and Dispersion Analysis of the Precise-Integration Time-Domain Method in Lossy Media

In this paper, both the numerical stability condition and dispersion relation of the precise-integration time-domain (PITD) method in lossy media are presented. It is found that the time step size of the PITD method is limited by both the spatial step size of the PITD method and the ratio of permittivity to conductivity. In numerical dispersion investigations, it is shown that: the numerical loss error of the PITD method is always positive; the numerical phase error of the PITD method can be positive or negative; the numerical loss and phase errors can be made nearly independent of the time step size; and as the spatial step size decreases, the amplitudes of the numerical loss and phase errors decrease. In good conductors, the numerical phase velocity of the PITD method is closer to the physical value as compared with the finite-difference time-domain method. The numerical phase anisotropy of the PITD method can be positive or negative. The numerical anisotropies of the PITD method in the 3-D case are usually larger than those in the 2-D case. There is a conductivity giving zero numerical phase anisotropy. These theoretical observations are confirmed by numerical experiments.

Alternating‐direction explicit algorithms for crank–nicolson‐based FDTD methods

AbstractThis article proposes a new unconditionally stable explicit finite‐difference time‐domain (FDTD) scheme for 3D Maxwell equation.The proposed scheme is based on alternating‐direction explicit method and Crank–Nicolson (CN)‐based FDTD method, and we present two versions of the proposed method. Numerical stability and dispersion analysis of the new algorithm were also presented. The results by the proposed method are compared with the results by the conventional FDTD method, alternating‐direction implicit method, and CN‐based FDTD method. As a result, it is confirmed that the proposed method is unconditionally stable and the efficiency is superior to the conventional method if the minimum cell size in the computational domains is required to be much smaller than the wavelength. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:1269–1273, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26802

High-order unified symplectic FDTD scheme for the metamaterials

A high-order unified symplectic finite-difference time-domain (US-FDTD) method, which is energy conserved, for modeling the metamaterials is proposed. The lossless Drude dispersive model is taken into account in US-FDTD scheme, and the detailed formulations of the proposed US-FDTD method are also provided. The high-order split perfectly matched layers (SPML) are used as the absorbing boundary conditions (ABCs) to terminate the computational domain. The analysis of Courant stability and numerical dispersion demonstrate that US-FDTD scheme is more efficient than the traditional time domain numerical methods. Focusing and refocusing of the electromagnetic wave in target detection is validated using the normal incident Gaussian beam with a matched slab. Oblique incidence results associated with the inverse Snell effect and the phase compensation effect of the composite slab further demonstrated the efficiency of the method. Numerical results for a more realistic structure are also included. All the results agree well with the theoretical prediction. The method proposed here can be directly put into using as a time-domain full-wave simulation tool for applications in metamaterials.

High-oder symplectic FDTD scheme for solving time-dependent Schrdinger equation

Using three-order symplectic integrators and fourth-order collocated spatial differences, a high-order symplectic finite-difference time-domain (SFDTD(3, 4)) scheme is proposed to solve the time-dependent Schrdinger equation. First, high-order symplectic framework for discretizing the Schrdinger equation is described. The numerical stability and dispersion analyses are provided for the FDTD(2, 2), FDTD(2, 4) and SFDTD(3, 4) schemes. The results are demonstrated in terms of theoretical analyses and numerical simulations. The spatial high-order collocated difference reduces the stability that can be improved by the high-order symplectic integrators. The SFDTD(3, 4) scheme and FDTD(2, 4) approach show better numerical dispersion than the traditional FDTD(2, 2) method. The simulation results of a two-dimensional quantum well and harmonic oscillator strongly confirm the advantages of the SFDTD(3, 4) scheme over the traditional FDTD(2, 2) method and other high-order approaches. The explicit SFDTD(3, 4) scheme, which is high-order-accurate and energy-conserving, is well suited for long-term simulation.

A NOVEL 3-D WEAKLY CONDITIONALLY STABLE FDTD ALGORITHM

For analyzing the electromagnetic problems with the flne structures in one or two directions, a novel weakly conditionally stable flnite-difierence time-domain (WCS-FDTD) algorithm is proposed. By dividing the 3-D Maxwell's equations into two parts, and applying the Crank-Nicolson (CN) scheme to each part, a four sub-step implicit procedures can be obtained. Then by adjusting the operational order of four sub-steps, a novel 3-D WCS-FDTD algorithm is derived. The proposed method only needs to solve four implicit equations, and the Courant{Friedrich{Levy (CFL) stability condition of the proposed algorithm is more relaxed and only determined by one space discretisation. In addition, numerical dispersion analysis demonstrates the numerical phase velocity error of the weakly conditionally stable scheme is less than that of the 3-D ADI-FDTD scheme.

Numerical analysis of a pollutant dispersion in subsurface soil

A hydraulic scale model was used in a laboratory experiment for the purpose of understanding the characteristics of a pollutant dispersion in subsurface soil. The experimental site was an area near a radioactive waste repository in the Republic of Korea. The hydraulic scale model was designed and manufactured in geometric similarity to the actual site. The tracer 99mTc which has a short half-life was injected instantaneously into the soil. Numerical simulations were performed to compare the measured radioisotope data and to investigate the overall dispersion patterns. The calculated concentrations are in good agreement with a time series of the measured concentrations at the midpoint of the detection lines.

Analysis of numerical dispersion properties of SFDTD and MRTD schemes

In this paper, the Maxwell's equations are written as Hamilton canonical equations by using Hamilton functional variation method. Maxwell's equations can be discretized with symplectic propagation technique combined with high-order difference schemes approximations to construct symplectic finite difference time domain (SFDTD) method. The high-order dispersion equations of the scheme for space is deduced. The numerical dispersion analysis is included, and it is compared with the multiresolution time-domain (MRTD) method based on the Daubechies scaling functions. Numerical results show high efficiency and accuracy of the SFDTD method.

Spurious solutions in mixed finite element method for Maxwell’s equations: Dispersion analysis and new basis functions

The finite element method is a well known computational technique used to obtain numerical solutions to boundary-value problems including Maxwell’s equations. This paper first presents a brief description of the mathematical structure, based on the De Rham diagram, to discretize Maxwell’s equations. Then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to evaluate the presence of spurious solutions for different basis functions. These unwanted spurious solutions appear when the same order of element is used for electric and magnetic fields, while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields. In this work, finite elements in both frequency and time domain are studied, and the effects of these spurious solutions in both domains are analyzed in one- and three-dimensional cases.

A 3-D LOD-FDTD Method for the Wideband Analysis of Optical Devices

An improved three-dimensional (3-D) locally one-dimensional finite-difference time-domain (LOD-FDTD) method is developed and applied to the wideband analysis of waveguide gratings. First, the formulation is presented, in which dispersion control parameters are introduced to reduce the numerical dispersion error and perfectly matched layers are simply implemented without the field components being split. Next, as a preliminary calculation, the wavelength response of the waveguide grating is analyzed in a two-dimensional problem. The dispersion control contributes to the accuracy improvement even with a large time step beyond the Courant-Friedrich-Levy limit. Finally, a 3-D waveguide grating is analyzed. The use of the dispersion control parameters only in the propagation direction enables us to employ a large time step for efficient calculations, i.e., the computation time can be reduced to about half that of the explicit counterpart. In the Appendix, maximum time step for providing a highly accurate result is also predicted using the numerical dispersion analysis.

Pseudospectral modeling and dispersion analysis of Rayleigh waves in viscoelastic media

Multichannel Analysis of Surface Waves (MASW) is one of the most widely used techniques in environmental and engineering geophysics to determine shear-wave velocities and dynamic properties, which is based on the elastic layered system theory. Wave propagation in the Earth, however, has been recognized as viscoelastic and the propagation of Rayleigh waves presents substantial differences in viscoelastic media as compared with elastic media. Therefore, it is necessary to carry out numerical simulation and dispersion analysis of Rayleigh waves in viscoelastic media to better understand Rayleigh-wave behaviors in the real world. We apply a pseudospectral method to the calculation of the spatial derivatives using a Chebyshev difference operator in the vertical direction and a Fourier difference operator in the horizontal direction based on the velocity–stress elastodynamic equations and relations of linear viscoelastic solids. This approach stretches the spatial discrete grid to have a minimum grid size near the free surface so that high accuracy and resolution are achieved at the free surface, which allows an effective incorporation of the free surface boundary conditions since the Chebyshev method is nonperiodic. We first use an elastic homogeneous half-space model to demonstrate the accuracy of the pseudospectral method comparing with the analytical solution, and verify the correctness of the numerical modeling results for a viscoelastic half-space comparing the phase velocities of Rayleigh wave between the theoretical values and the dispersive image generated by high-resolution linear Radon transform. We then simulate three types of two-layer models to analyze dispersive-energy characteristics for near-surface applications. Results demonstrate that the phase velocity of Rayleigh waves in viscoelastic media is relatively higher than in elastic media and the fundamental mode increases by 10–16% when the frequency is above 10 Hz due to the velocity dispersion of P and S waves.

Numerical wave propagation for the triangular P1 DG– P2 finite element pair

The f-plane and β-plane wave propagation properties are examined for discretisations of the linearised rotating shallow-water equations using the P1 DG – P2 finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the f-plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the P1 DG – P2 finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f, and which do not propagate. A restriction of the P1 DG velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised P1 DG – P2 equations in the β-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.

Numerical Dispersion Analysis of the Unconditionally Stable Three-Dimensional LOD-FDTD Method

The numerical dispersion characteristics of the recently developed three-dimensional unconditionally stable locally- one-dimensional finite-difference time-domain (LOD-FDTD) method are derived analytically. The effect of grid size and the Courant Friedrich Lewy (CFL) limits on dispersion are studied in detail. The LOD-FDTD method allows larger time steps as compared to the conventional FDTD method (CFL limit). The analysis shows that the unconditionally stable three-dimensional LOD-FDTD method has an advantage over the conventional FDTD method when modeling structures that require fine grids. The LOD-FDTD method allows larger CFL numbers as long as the dispersion error remains in acceptable range.

Experimental and numerical analysis of the jet dispersion from a bent chimney around an obstacle

An experimental study and a numerical modelling analysis were carried out simultaneously to study the flow field structure issuing from a chimney around an obstacle. The main purpose of this study is to evaluate the impact of the jet emitted from a chimney (bent or straight) on the dynamics and the turbulent features of the surrounding flow. The consideration of these features is particularly pertinent to the understanding of mixing between the interacting flows which may be very important in controlling pollutant dispersion in the atmosphere. The experimental data are depicted by means of a PIV technique; whereas the numerical three-dimensional model is simulated through the resolution of the different governing Navier–Stokes equations. The volume finite method, together with the second order turbulent closure model (RSM), was adopted. Variations in obstacle form (cylindrical or parallelepiped) and chimney configuration (bent or straight) were tested and features studied were: the global jet plume, the windward and leeward jet spread; the size, location and magnitude of the reverse flow region; the penetration and the deflection of the jet trajectory around the obstacle. All these considerations allowed us to characterize well the impact of the injection of the jet emitted from the chimney within the crossflow, and its spreading around the obstacle and within the whole domain. Such characterization is very important with regard to pollutant dispersion and consequently to the environmental impact. Indeed, the different species contained within the emitted fumes are mainly directed by the velocity components and their mixing and progression within the domain and around the obstacle are closely related.

The ADI-FDTD Method for Transverse-Magnetic Waves in Conductive Materials

A numerical dispersion analysis of the alternating-direction implicit finite-difference time-domain method is introduced for transverse-magnetic waves in lossy materials. To provide a general study, the conduction terms are discretized by using a weighted average in time. It is found that the numerical dispersion relation is different to that of the transverse-electric (TEz) case. Moreover, contrary to what happens in the TEz case, no set of weighted-average parameters makes the accuracy of the formulation independent of how well the time step resolves the material relaxation-time constant. To overcome this problem, a split-field version of Maxwell's equations is considered as the governing equations. Then, by synchronizing in time the lossy terms with the spatial derivatives, the resulting formulation has the same numerical dispersion relation as in the TEz case. Moreover, its accuracy becomes independent of the quality of the resolution of the relaxation-time constant by the time step.

Long time stable compact fourth-order scheme for time domain Maxwell's equations

A compact staggered backward differentiation method (CSBD(4,4)) is proposed. Theoretical analyses of numerical stability and dispersion are straightforward, and comparisons with the staggered backward differentiation method (SBD(4,4)) are provided. It shows that the numerical dispersion of the proposed scheme is greatly reduced.

A Compact Symplectic High-Order Scheme for Time-Domain Maxwell's Equations

A fourth-order compact symplectic finite-difference time-domain (CS-FDTD) method for modeling long-range propagation is proposed. Theoretical analyses of numerical stability and dispersion are presented, and the comparisons to Fang's high-order FDTD and symplectic FDTD (S-FDTD) method are provided. One-dimensional (1-D) numerical simulation is performed to investigate the distortion of the long-range pulse propagation. It indicates the improved performance of the CS-FDTD approach compared to the S-FDTD method.

Analysis of Two Alternative ADI-FDTD Formulations for Transverse-Electric Waves in Lossy Materials

A numerical dispersion analysis of the alternating-direction implicit finite-difference time-domain method for transverse-electric waves in lossy materials is presented. Two different finite-difference approximations for the conduction terms are considered: the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">double-average</i> and the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">synchronized</i> schemes. The numerical dispersion relation is derived in a closed form and validated through numerical simulations. This study shows that, despite its popularity, the accuracy of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">double-average</i> scheme is sensitive to how well the relaxation-time constant of the material is resolved by the time step. Poor resolutions lead to unacceptably large numerical errors. On the other hand, for good conductors, the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">synchronized</i> scheme allows stability factors as large as 100 to be used without deteriorating the accuracy significantly.

Determination of the Voigt constant of phthalocyanines by magneto-optical Kerr-effect spectroscopy

Thin films of organic molecular semiconductors from the class of phthalocyanines are investigated by means of magneto-optical Kerr effect (MOKE) spectroscopy at room temperature. A numerical analysis of the energy dispersion of the real and imaginary part of the complex magneto-optical Kerr rotation angle in the visible to near ultraviolet spectral range allows the first determination of the magneto-optical material constant, the so-called Voigt constant, for a molecular material and the results are compared to density-functional theory calculations. The Voigt constant is only 1--2 orders of magnitude smaller compared with soft ferromagnetic materials such as Ni. The amplitude of the Voigt constant is exploited to determine the molecular orientation in thin films with high accuracy.

Numerical analysis of release, dispersion and combustion of liquid hydrogen in a mock-up hydrogen refuelling station

Computational Fluid Dynamics (CFD) simulations of hydrogen dispersion and combustion in accident scenarios in a mock-up liquid hydrogen (LH 2) refuelling station have been performed with the codes ADREA-HF and REACFLOW. The assumed accident scenarios are caused by a hose break during LH 2 car refuelling. Two main configurations have been taken into account: a mitigated case and a non-mitigated case. For each configuration, 5 different ambient conditions are assumed: a case with no wind at all and four cases with a 5 m/s wind speed, coming from the south, north, east and west direction. The numerical simulations show the effect of the wind on the dispersion and explosion of the hydrogen flammable cloud. The effect of ignition position has also been investigated. The expectation that the wind plays a positive role in increasing the dispersion of the flammable cloud and therefore in reducing the overpressures in case of explosions does not hold true for all wind directions. Depending on the wind direction and on the station layout, the wind can have the effect of trapping the flammable cloud with more reactive concentrations and higher level of turbulence within the station, generating a similar level of overpressure as in the stagnant case although the amount of flammable mixture is much smaller than in the stagnant case.

ADI-FDTD Method With Fourth Order Accuracy in Time

This letter presents an unconditionally stable alternating direction implicit finite-difference time-domain (ADI-FDTD) method with fourth order accuracy in time. Analytical proof of unconditional stability and detailed analysis of numerical dispersion are presented. Compared to second order ADI-FDTD and six-steps SS-FDTD, the fourth order ADI-FDTD generally achieves lower phase velocity error for sufficiently fine mesh. Using finer mesh gridding also reduces the phase velocity error floor, which dictates the accuracy limit due to spatial discretization errors when the time step size is reduced further.