Numerical Dispersion Analysis Research Articles

Application of spatial filtering high order FDTD method in the multi-physics field coupled equations

The traditional finite-difference time-domain method (FDTD(2, 2)) is widely employed to solve the coupled equations of quantum-electromagnetic multi-physics. However, the time step of the traditional explicit FDTD(2, 2) method is limited by the Courant-Friedrichs-Lewy (CFL) condition. Besides, the FDTD(2, 2) method only achieves the second-order accuracy, which can result in significant error accumulation. To overcome this problem, we propose the SF-HO-FDTD(2, q) method with scalable time stability conditions for solving the multi-physics field coupled equations (q denotes q-th order spatial difference scheme). This approach integrates spatial filtering (SF) and higher-order FDTD (HO-FDTD) methods. Moreover, the proposed method requires no further derivation of the iterative equations of the traditional HO-FDTD(2, q) method. It only needs to introduce the SF operation in each numerical iteration process to filter out the spatial high-frequency components arising from the use of time steps that do not satisfy the CFL stability condition. This condition is applied to ensure the stability of the numerical method. Therefore, the proposed method has a high compatibility with the traditional HO-FDTD(2, q) method. Then, the numerical dispersion analysis of the SF-HO-FDTD(2, q) method is theoretically studied. Finally, numerical examples are carried out to confirm the accuracy and efficiency of the strategy suggested in this study.

Finite-difference frequency-domain method with QR-decomposition-based complex-valued adaptive coefficients for 3D diffusive viscous wave modelling

Abstract The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC.

Complex-valued adaptive-coefficient finite-difference frequency-domain method for wavefield modeling based on the diffusive-viscous wave equation

The diffusive-viscous wave (DVW) equation is an effective model for analyzing seismic low-frequency anomalies and attenuation in porous media. To effectively simulate DVW wavefields, the finite-difference or finite-element method in the time domain is favored, but the time-domain approach proves less efficient with multiple shots or a few frequency components. The finite-difference frequency-domain (FDFD) method featuring optimal or adaptive coefficients is favored in seismic simulations due to its high efficiency. Initially, we develop a real-valued adaptive-coefficient (RVAC) FDFD method for the DVW equation, which ignores the numerical attenuation error and is a generalization of the acoustic adaptive-coefficient FDFD method. To reduce the numerical attenuation error of the RVAC FDFD method, we introduce a complex-valued adaptive-coefficient (CVAC) FDFD method for the DVW equation. The CVAC FDFD method is constructed by incorporating correction terms into the conventional second-order FDFD method. The adaptive coefficients are related to the spatial sampling ratio, number of spatial grid points per wavelength, and diffusive and viscous attenuation coefficients in the DVW equation. Numerical dispersion and attenuation analysis confirm that, with a maximum dispersion error of 1% and a maximum attenuation error of 10%, the CVAC FDFD method only necessitates 2.5 spatial grid points per wavelength. Compared with the RVAC FDFD method, the CVAC FDFD method exhibits enhanced capability in suppressing the numerical attenuation during anelastic wavefield modeling. To validate the accuracy of our method, we develop an analytical solution for the DVW equation in a homogeneous medium. Three numerical examples substantiate the high accuracy of the CVAC FDFD method when using a small number of spatial grid points per wavelength, and this method demands computational time and computer memory similar to those required by the conventional second-order FDFD method. A fluid-saturated model featuring various layer thicknesses is used to characterize the propagation characteristics of DVW.

Numerical analysis of vapor dispersion from compressed hydrogen (H2) storage vessels

The rising demand for hydrogen fuel, driven by the emergence of fuel cell electric vehicles, underscores the need to optimize refueling station efficiency and affordability while prioritizing safety and performance. Compressed gas hydrogen storage emerges as a practical solution however safety across production, storage, and distribution is paramount for broader acceptance of hydrogen technologies. Any incidents could undermine public trust, emphasizing the importance of mitigating risks such as hydrogen leakage. This study investigates hydrogen dispersion and conducts consequence analyses for potential hazards, considering stability, ambient temperature, wind speed, and process parameters like vessel temperature, pressure, and leakage diameter. It assesses various scenarios, including high-pressure storage vessels and generic refueling station layouts, by employing integral models of ALOHA, PHAST and HyRAM. Findings showed that process parameters significantly influence hazard severity, with leakage diameter having a notable impact. Common safety vulnerabilities in fuel cell vehicles and refueling stations are highlighted, emphasizing adherence to international regulations and standards for enhanced safety protocols.

Numerical Dispersion Analysis of the One-Dimensional Iterated Crank-Nicolson-Based FDTD Method

Numerical Dispersion Analysis of the One-Dimensional Iterated Crank-Nicolson-Based FDTD Method

An FCC-FDTD Method on Nonuniform Grids With Guaranteed Stability for Electromagnetic Analysis

The finite-difference time-domain (FDTD) method based on nonuniform face-centered cubic (FCC) grids, termed as NU-FCC-FDTD method, is proposed in this article. The FCC-FDTD method with uniform grids has a smaller numerical dispersion error than the traditional FDTD methods. To further enhance its efficiency and scalability, the NU-FCC-FDTD method is proposed to solve complex electromagnetic problems. Unlike the traditional FDTD methods based on Yee’s grids, for which the nonuniform grid technique is relatively straightforward, instability may occur when trying to apply nonuniform grids into the FCC-FDTD method. To ensure stability, cell-size-based correction factors are carefully defined in the time-marching formulations after taking cell sizes into consideration. Through comprehensive numerical dispersion analysis, the NU-FCC-FDTD method shows numerical dispersion advantage over the FDTD methods and the FCC-FDTD method with uniform grids. Furthermore, the stability condition is also given by using the parameter scanning method and verified in numerical examples. Four examples, including electromagnetic wave propagation, a dielectric rod (DR) resonator, a composite metallic-dielectric radome and the electromagnetic energy absorption of human head, are carried out to demonstrate its performance. Results show that the proposed NU-FCC-FDTD method can significantly improve efficiency compared with that of the FCC-FDTD method without compromising accuracy.

Analysis of Stability and Numerical Dispersion of 3-D HIE-FDTD Method Including Lumped Elements

Analysis of Stability and Numerical Dispersion of 3-D HIE-FDTD Method Including Lumped Elements

An FDTD method for multiterm modified Lorentz media and comparisons with existing schemes

AbstractA finite‐difference time‐domain (FDTD) method for modified Lorentz (mLor) media is proposed based on the order reduction of the governing differential equations. The introduced algorithm is verified in the benchmark problem of wave propagation through a thin slab. Several FDTD methods for the modeling of mLor media are also discussed and a numerical dispersion analysis is conducted. Comparisons in terms of accuracy and numerical simulations prove the validity of the schemes for the study of optics and photonic applications.

Seamless numerical analysis of transient infectious droplet dispersion and inhalation exposure - In silico study

To address the problem of airborne transmission caused by droplets/droplet nuclei containing infectious viruses such as influenza A, tuberculosis, and SARS-CoV-2, the transmission dynamics of infectious droplets in indoor environments, from the generation/emission from an infected subject to the exposure of the target subject via indoor air, must be comprehended seamlessly and accurately. Research on indoor airborne transmission requires rigorous prior ethical review, and experimental studies on humans (in vivo and/or in vitro) are severely limited. Therefore, research methods based on in silico models; i.e., numerical modeling and analysis approaches, are preferred, owing to their flexibility for case and sensitivity analyses. In this study, assuming airborne transmission of SARS-CoV-2 in an indoor environment, we develop and perform a seamless and continuous numerical analysis of infectious droplet dispersion from a coughing infected subject and the subsequent inhalation exposure of a target subject (via the respiratory tract) through transient breathing. We focus on the effects of respiration patterns, particle-size changes due to evaporation, and indoor physical distance between the subjects, on the exposure concentration and distribution of inhaled droplets in the respiratory tract. The results show that, when an infected person coughs within a physical distance of 1 m, inhalation exposure results in significant differences in the inhalation–exhalation timing of breathing, and local environmental conditions have a dominant effect on the inhalation exposure. In short-range airborne transmission, the particle-size change due to evaporation has a dominant effect on the estimation of the inhalation exposure concentration in the respiratory tract.

Numerical Stability and Dispersion Analysis of the 2-D FDTD Method Including Lumped Elements

The numerical stability and dispersion analysis of the extended two-dimensional finite-difference time-domain (2D-FDTD) method are systematically studied. Particularly, three different passive linear lumped elements including the resistor, inductor, and capacitor are analyzed, respectively. Moreover, three different formulations of the explicit, semi-implicit, and implicit schemes are discussed, respectively. Furthermore, by combining the von Neumann technique and Jury criterion, the numerical stability of the extended 2D-FDTD method is analyzed, which has not been reported thus far. Theoretical results show that: 1) For the resistor, the stability condition is same as the FDTD method unloaded case. 2) For the inductor, in the explicit and implicit schemes, the stability is connected with the value of the inductance; for the semi-implicit scheme, the stability is independent of the value of the inductance. 3) For the capacitor, the stability relationship is related to both the mesh size and the value of the capacitance. On the other hand, based on the Norton equivalent circuit, the analysis of the numerical dispersion of the extended 2D-FDTD is presented; and some interesting theoretical results are deduced. Finally, the microstrip circuits including the three lumped elements are simulated to demonstrate the validity of the theoretical results.

Attenuation-compensated reverse time migration based on the stereo-modeling operator

The viscosity in the subsurface is ubiquitous. Prestack reverse time migration (RTM) based on the viscoacoustic wave equation is an accurate imaging method to compensate attenuation. The resolution of an imaging result highly depends on the accuracy of the wave equation solution. We extend the nearly analytic center difference (NACD) method to a viscoacoustic wave equation to compensate seismic attenuation. The NACD method can achieve fourth-order accuracy in the time and space domain. We use the pressure and its gradients to approximate the spatial derivatives based on stereo modeling. Numerical simulations and dispersion analysis demonstrate that our NACD has less numerical dispersion and is more accurate than the Lax-Wendroff correction (LWC) method. Thus, the NACD is less expensive in terms of computational time and storage space. Numerical simulations indicate that the forward modeling of the viscoacoustic wavefield via the NACD is able to obtain more accurate wavefields compared with the conventional LWC method and can accurately simulate the effect of the viscosity on waves in attenuative media. When applied to RTM, the viscous effects of the subsurface can be compensated. Synthetic examples demonstrate that the attenuation-compensated RTM using NACD can obtain an image with higher resolution in attenuative media compared with that using the acoustic RTM. Furthermore, the image determined by the NACD is clearer than that of the LWC method in RTM.

A Simplified Novel Link for A Simplified Stability Analysis of Finite Difference Time Domain Method

Numerical stability and numerical dispersion analyses are critical subjects for Finite Difference Time Domain (FDTD) method. To perform these analyses, first of all, an equivalency of the FDTD numerical dispersion equation for Maxwell’s equations and wave equation is proven in this study. Then, based on those calculations, a simplified version of a novel link is developed. Using this simplified version, a stability criterion and an amplification factor of the FDTD method are more easily extracted. Therefore, the FDTD stability analysis becomes simpler. The theoretical findings are validated by a numerical example of a late time simulation interval in the FDTD method. In particular, the effect of a hard FDTD source and a soft FDTD source on the growth (amplification) factor is also investigated.

Cloaking Rayleigh waves via symmetrized elastic tensors

In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media. They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation x=Ξ(X), upon choosing the specific kinematic relation U(Ξ(X))=u(x) between displacement fields in virtual, i.e. reference, (U) and transformed, i.e. cloaked, (u) domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking. In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step towards the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.

Numerical analysis of vortex formation and particle dispersion in a supersonic compressible particle-laden mixing layer

Numerical analysis of vortex formation and particle dispersion in a supersonic compressible particle-laden mixing layer

Design and numerical analysis of large negative dispersion and ultra-high nonlinearity $$CS_{2}$$ filled LCPCF

Design and numerical analysis of large negative dispersion and ultra-high nonlinearity $$CS_{2}$$ filled LCPCF

Eliminating finite-grid instabilities in gyrokinetic particle-in-cell simulations

Finite-grid (aliasing) instabilities are known to place severe limitations on momentum-conserving particle-in-cell (PIC) methods applied to models including charge separation effects by requiring the resolution of the Debye length. Gyrokinetic models, on the other hand, generally enforce quasi-neutrality thereby removing the Debye length analytically. Recent studies with momentum-conserving PIC applied to gyrokinetic models, however, show that a manifestation of this instability exists in certain physical parameter regimes for arbitrary spatial resolution. In this paper, we show that a simple reformulation of the discrete equations, making use of a co-located discretization of the continuity equation, eliminates this instability for all practical purposes. We perform numerical dispersion analyses for both the original and reformulated schemes, including the effects of finite mean parallel velocity and finite β. We demonstrate that the reformulated scheme is numerically stable for stationary plasmas at any spatial resolution and for plasmas in which the electron mean parallel velocity is smaller than the electron thermal velocity (generally ensured by ambipolarity in plasmas of interest). This reformulation may be particularly useful for codes with complicated meshes, where high-order shape functions or energy-conserving schemes are difficult to implement. The analysis presented here may also be useful for explaining the success of previously considered methods to remove particle instabilities, such as the split-weight scheme.

Numerical and experimental analysis of Lagrangian dispersion in two-dimensional chaotic flows.

We present a review and a new assessment of the Lagrangian dispersion properties of a 2D model of chaotic advection and diffusion in a regular lattice of non stationary kinematic eddies. This model represents an ideal case for which it is possible to analyze the same system from three different perspectives: theory, modelling and experiments. At this regard, we examine absolute and relative Lagrangian dispersion for a kinematic flow, a hydrodynamic model (Delft3D), and a laboratory experiment, in terms of established dynamical system techniques, such as the measure of (Lagrangian) finite-scale Lyapunov exponents (FSLE). The new main results concern: (i) an experimental verification of the scale-dependent dispersion properties of the chaotic advection and diffusion model here considered; (ii) a qualitative and quantitative assessment of the hydro-dynamical Lagrangian simulations. The latter, even though obtained for an idealized open flow configuration, contributes to the overall validation of the computational features of the Delft3D model.

Numerical dispersion and dissipation of the triangle-based discontinuous Galerkin method for acoustic and elastic velocity-stress equations

In this work, we present the numerical dispersion and dissipation analyses of the triangle-based discontinuous Galerkin method (DGM) for acoustic and elastic velocity-stress equations. The analysis is based on the eigenvalue problem. The applicability of the flux-based DGM is affected by many factors, including numerical flux, basis function, mesh element, and time-stepping scheme. Based on the semi-discrete analysis, we compare different numerical fluxes—Centred flux, local Lax-Friedrichs (LLF) flux, and Godunov flux, different basis functions—Koornwinder-Dubiner (KD) polynomial, Legendre polynomial, and simple monomial function, and four types of triangular meshes. The fully discrete schemes include the third-order total variation diminishing Runge-Kutta (TVD RK) method and the third-order weighted Runge–Kutta (WRK) method. The results indicate that numerical dispersion and dissipation behave differently for different numerical fluxes and mesh types, which verifies the importance of considering numerical fluxes and mesh types. We find that using LLF and Godunov fluxes, and the mesh pattern of case 3 has attractive advantages for numerical simulation. The result of the dispersion-dissipation analysis with different basis functions demonstrates that for high-order integral accuracy, three kinds of basis functions have the same performance in suppressing numerical dispersion and dissipation in triangular elements. The fully discrete dispersion–dissipation analysis shows that the time-stepping scheme introduces numerical dispersion and dissipation and the fully discrete case with a small sampling ratio and Courant number has a better effect in suppressing numerical dispersion and dissipation. Finally, we provide several numerical experiments to validate our theoretical finding, and numerical results are consistent with theoretical dispersion-dissipation analyses.

Design of Ultra-Compact On-Chip Discrete Phase Filters for Broadband Dispersion Management

This paper provides an in-depth theoretical and numerical analysis of a group-velocity dispersion (GVD) management scheme for broadband waveforms using ultra-compact on-chip discrete spectral phase filters based upon waveguide Bragg gratings (WBGs) in a silicon-on-insulator (SOI) platform. Through this technique, mm-long discrete phase filters can be designed to impart a target arbitrary GVD profile on a high-rate (typically, $\geq\!10$ GHz) periodic pulsed waveform, including fully customized and extremely large second and higher-order dispersion terms (e.g., equivalent to 10,000 km of a standard single-mode fiber), over a broad frequency bandwidth (up to $\sim\!3$ THz, demonstrated here). The capabilities and limitations of this technique to impart a target GVD profile over an arbitrary (generally, non-periodic) broadband signal are also studied. We show that customized GVD lines can be efficiently implemented offering a net group-delay excursion in the hundreds-of-ps range using mm-long integrated discrete phase filters. Additionally, we suggest and numerically demonstrate a simple and practical strategy to improve significantly the performance of the discrete phase filtering approach for application on non-periodic waveforms, by combining the discrete phase filter with a suitable periodic resonance (frequency-comb) amplitude filter. An extensive tolerance analysis is conducted, and we conclude that the proposed SOI design framework is well within practical fabrication requirements as well as robust to the expected variability in the main WBG device design parameters.

Numerical analysis of dispersion and self-phase modulation for elimination of chaotic time delay

We mainly investigate the influence of group velocity dispersion (GVD) and self-phase modulation (SPM) in single-mode fiber on chaotic time delay signature (TDS) of single feedback semiconductor laser. In the simulation, the second-order dispersion coefficient and the average input optical power in fibers are adjusted to control GVD and SPM. The results show that after the fiber length reaches the dispersion length of the chaotic laser, the chaotic TDS caused by single optical feedback begins to be suppressed due to the significant enhancement of GVD, and then the value of the autocorrelation coefficient at the chaotic TDS descends at the fastest suppression rate of 0.5 dB/km. When chaos laser propagates about 20 km in the fibers, duo to the accumulation of SPM to the significant range of action, the chaotic TDS achieves the best stable suppression state, and the suppression effect can reach 11.6 dB.