Pseudospectral Time-domain Algorithm Research Articles

Grid-adaptive Fourier pseudospectral time domain model for the light scattering simulation of atmospheric nonspherical particles.

PSTD (pseudospectral time domain) is recognized as one of the powerful models to accurately calculate the scattering properties of nonspherical particles. But it is only good at the computation in coarse spatial resolution, and large "staircase approximation error" will occur in the actual computation. To solve this problem, the variable dimension scheme is introduced to improve the PSTD computation, in which, the finer grid cells are set near the particle's surface. In order to ensure that the PSTD algorithm can be performed on non-uniform grids, we have improved the PSTD with the space mapping technique so that the FFT algorithm can be implemented. The performance of the improved PSTD (called "IPSTD" in this paper) is investigated from two aspects: for the calculation accuracy, the phase matrices calculated by IPSTD are compared with those well tested scattering models like Lorenz-Mie theory, T-matrix method and DDSCAT; for computational efficiency, the computational time of PSTD and IPSTD are compared for the spheres with different sizes. From the results, it can be found that, the IPSTD scheme can improve the simulation accuracy of phase matrix elements notably, especially in the large scattering angles; though the computational burden of IPSTD is larger than that of PSTD, its computational burden does not increase substantially.

An Adaptive High-Order Transient Algorithm to Solve Large-Scale Anisotropic Maxwell’s Equations

This article presents a stabilized nodal discontinuous Galerkin pseudospectral time-domain (DG-PSTD) algorithm for fully anisotropic electromagnetic waves. This solver permits arbitrary high-order basis functions and adaptive hexahedral elements, thus very efficient for large-scale wave propagation in complex media. Maxwell’s equations are reformulated in a unified hyperbolic form, where a localized anisotropic Riemann solver is derived to serve as a numerical flux to exchange information across adjacent elements in the DG-PSTD scheme. This local analysis method also helps impose the time-domain anisotropic plane wave incidence in the total/scattering field framework. Numerical validations and applications demonstrate the efficiency, accuracy, and capability of this new high-order solver for 3-D large-scale generally anisotropic electromagnetic media.

A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm

We present a solver of 3D two-fluid plasma model for the simulation of short-pulse laser interactions with plasma. This solver resolves the equations of the two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD) method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler equations with source term: while Euler equations are solved with a composite scheme mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against several well-known problems of plasma physics. Numerical simulations gave results in excellent agreement with analytical solutions or with previous results from the literature.

Stabilized DG-PSTD Method With Nonconformal Meshes for Electromagnetic Waves

We present a node-based discontinuous Galerkin (DG) pseudospectral time domain (PSTD) algorithm, with adaptive nonconformal unstructured meshes, for 3-D large-scale Maxwell’s equations. This algorithm is a combination of a new DG algorithm and a PSTD method, where spectral accuracy is achieved via the PSTD algorithm, while the DG serves as a stable coupling for multiple domains with unstructured hexahedra. Time marching is efficient because the mass matrix in the DG-PSTD algorithm is exactly diagonal. The scheme is low-storage and scalable because the stiffness matrix is localized into a small shared matrix. Furthermore, arbitrary nonconformal meshes can be adaptively realized, increasing the flexibility of complex media modeling. Our numerical results corroborate the long-time stability, high efficiency, and high-order accuracy of the proposed solver. Finally, an adaptive application of 5G electromagnetic signal propagation demonstrates the efficiency and capability of the proposed high-order solver.

GPU acceleration of time gating based reverse time migration using the pseudospectral time-domain algorithm

We present a Graphics Processing Units (GPU) implementation of time gating based reverse time migration (TG-RTM) which uses the pseudospectral time-domain (PSTD) algorithm to solve the acoustic wave equation. TG-RTM adopts the prior information of surrounding media to strengthen the correlation between the wavefields, thus has advantages in locating the targets over traditional reverse time migration (RTM) methods. The PSTD algorithm adopts fast Fourier transform (FFT) to obtain the spatial derivatives and a perfectly matched layer as an absorbing boundary condition to eliminate the wraparound effect introduced by the FFT periodicity assumption. Under the Nyquist sampling theorem, the spatial sampling density of the PSTD algorithm requires only two points per minimum wavelength. Thus, the PSTD algorithm can solve the time dependent partial differential equations efficiently and save mass computer memory. Compared with traditional RTM based on the finite difference time domain (FDTD) algorithm, the proposed RTM based on the PSTD algorithm can be implemented on a memory-limited GPU and can solve much larger models. To secure a better performance and generality of FFT in GPU, we present a scheme which combines 1D FFT with matrix transpositions instead of using 3D FFT directly. The matrix transpositions use shared memory to improve memory access efficiency. We also apply an efficient FFT scheme which replaces even-sized R2C FFT with a half-sized C2C FFT. For a small amount and balanced memory swapping from computer to GPU, we save the boundaries in lieu of checkpointing scheme when we propagate the source wavefield forward and backward. The proposed RTM has an acceleration ratio of about 80 times by a Tesla K20X GPU card on a desktop computer. The simulation results of 2D and 3D models demonstrate that the proposed RTM is fast and inexpensive.

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

An exact Riemann solver for wave propagation in arbitrary anisotropic elastic media with fluid coupling

We present a nonconformal mesh discontinuous Galerkin pseudospectral time domain algorithm for arbitrary anisotropic elastic/acoustic wave propagation problems. An exact Riemann solver is compactly derived to resolve the accurate coupling of multiple domains in the discontinuous Galerkin framework, including heterogeneous anisotropic solid–solid, acoustic–acoustic, and anisotropic solid–fluid interactions. We simplify the eigenvalue problem in the Riemann solution from the rank of 9 to 3, and introduce the generalized wave impedance with more physical insight. Validations and verifications with independent codes and analytical solutions illustrate the accuracy, flexibility, and stability of our algorithm.

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

Isotropic Riemann Solver for a Nonconformal Discontinuous Galerkin Pseudospectral Time-Domain Algorithm

We present a discontinuous Galerkin pseudospectral time-domain (DG-PSTD) algorithm to solve elastic-/acoustic-wave propagation problems. The developed DG-PSTD algorithm combines the merits of flexibility from a finite-element method and spectral accuracy and efficiency from a high-order pseudospectral method, while having a flavor closer to a finite-volume method. This numerical approach not only uses structured/unstructured conformal meshes but also handles nonconformal meshes (h-adaptivity) with nonuniform approximation orders (p-adaptivity) in different regions, thus leading to high flexibility and efficiency for heterogeneous multiscale problems. To implement the discontinuous Galerkin algorithm, a concise but more general heterogeneous Riemann solver is provided to effectively and accurately resolve the coupling of multiple subdomains for both elastic–elastic/fluid–fluid and fluid–solid coupling. Finally, numerical results demonstrate the flexibility, high accuracy, and efficiency of our method for elastic-/acoustic-wave simulation.

Reverse Time Migration Using the Pseudospectral Time-Domain Algorithm

We propose a pre-stack reverse time migration (RTM) seismic imaging method using the pseudospectral time-domain (PSTD) algorithm. Traditional pseudospectral method uses the fast Fourier transform (FFT) algorithm to calculate the spatial derivatives, but is limited by the wraparound effect due to the periodicity assumed in the FFT. The PSTD algorithm combines the pseudospectral method with a perfectly matched layer (PML) for acoustic waves. PML is a highly effective absorbing boundary condition that can eliminate the wraparound effect. It enables a wide application of the pseudospectral method to complex models. RTM based on the PSTD algorithm has advantages in the computational efficiency compared to traditional methods such as the second-order and high order finite difference time-domain (FDTD) methods. In this work, we implement the PSTD algorithm for acoustic wave equation based RTM. By applying the PSTD-RTM method to various seismic models and comparing it with RTM based on the eighth-order FDTD method, we find that PSTD-RTM method has better performance and saves more than 50% memory. The method is suitable for parallel computation, and has been accelerated by general purpose graphics processing unit.

3D pseudospectral time domain for modeling second-harmonic generation in periodically poled lithium niobate ridge-type waveguides

We report an application of the tri-dimensional pseudo-spectral time domain algorithm, that solves with accuracy the nonlinear Maxwell's equations, to predict second harmonic generation in lithium niobate ridge-type waveguides with high index contrast. Characteristics of the nonlinear process such as conversion efficiency as well as impact of the multimode character of the waveguide are investigated as a function of the waveguide geometry in uniformly and periodically poled medium.

Dispersive multidomain pseudospectral time‐domain method for full‐wave analysis of radio‐frequency invisible cloaks

To implement full-wave simulation of electromagnetic wave impinging invisible cloaks, an efficient multidomain pseudospectral time-domain algorithm is developed in this study. The proposed method solves the same forms of Maxwell's equations in both the regular and cloaking areas, and then applies different methods to update the electric and magnetic field components, respectively. The new method requires less grid points per wavelength, meanwhile, the multidomain strategy relieves the complicated treatment of electric and magnetic field components based on Yee grids within finite-difference time-domain methods, and shows advantage of dealing with internal inhomogeneities within the cloaking materials. The results validated feasibility and efficiency of the proposed method in left-hand materials.

Pseudo-Spectral simulation of 1D nonlinear propagation in heterogeneous elastic media

In this work, a 1D Pseudo-Spectral Time Domain (PSTD) algorithm has been developed for solving elastic wave equation in nonlinear heterogeneous solids using FFTs for calculation of the spatial differential operator on staggered grid. The solver uses a staggered fourth order Adams–Bashforth method, by which stress and particle velocity are updated at alternating half time steps, to integrate forward in time. To circumvent wraparound inherent to FFT-based pseudo-spectral simulation, Convolution Perfectly Matched Layer (CPML) boundary condition has been used to eliminate implementation problems linked in classical PML to the introduction in nonlinear elasticity of a time dependent bulk modulus. Different kinds of nonlinear elastic models (quadratic and cubic nonlinearity, Nazarov hysteretic nonlinearity, bi-modular nonlinearity, PM-Space nonlinearity) have been implemented. The present study will focus on the comparison of nonlinear signature (harmonics generation, shock, frequency shift and attenuation) of these different kinds of nonlinearity for rod resonance, shock wave generation. These results are expected to be useful in helping to determine the predominant nonlinear mechanism in a specific experiment.

2-D PSTD Simulation of the time-reversed ultrasound-encoded deep-tissue imaging technique

We present a robust simulation technique to model the time-reversed ultrasonically encoded (TRUE) technique for deep-tissue imaging. The pseudospectral time-domain (PSTD) algorithm is employed to rigorously model the electromagnetic wave interaction of light propagating through a macroscopic scattering medium. Based upon numerical solutions of Maxwell's equations, the amplitude and phase are accurately accounted for to analyze factors that affect the TRUE propagation of light through scattering media. More generally, we demonstrate the feasibility of modeling light propagation through a virtual tissue model of macroscopic dimensions with numerical solutions of Maxwell's equations.

Numerical absorbing boundary conditions based on a damped wave equation for pseudospectral time-domain acoustic simulations.

In the context of wave-like phenomena, Fourier pseudospectral time-domain (PSTD) algorithms are some of the most efficient time-domain numerical methods for engineering applications. One important drawback of these methods is the so-called Gibbs phenomenon. This error can be avoided by using absorbing boundary conditions (ABC) at the end of the simulations. However, there is an important lack of ABC using a PSTD methods on a wave equation. In this paper, we present an ABC model based on a PSTD damped wave equation with an absorption parameter that depends on the position. Some examples of optimum variation profiles are studied analytically and numerically. Finally, the results of this model are also compared to another ABC model based on an hybrid formulation of the scalar perfectly matched layer.

Numerical stability analysis of the pseudo-spectral analytical time-domain PIC algorithm

The pseudo-spectral analytical time-domain (PSATD) particle-in-cell (PIC) algorithm solves the vacuum Maxwellʼs equations exactly, has no Courant time-step limit (as conventionally defined), and offers substantial flexibility in plasma and particle beam simulations. It is, however, not free of the usual numerical instabilities, including the numerical Cherenkov instability, when applied to relativistic beam simulations. This paper derives and solves the numerical dispersion relation for the PSATD algorithm and compares the results with corresponding behavior of the more conventional pseudo-spectral time-domain (PSTD) and finite difference time-domain (FDTD) algorithms. In general, PSATD offers superior stability properties over a reasonable range of time steps. More importantly, one version of the PSATD algorithm, when combined with digital filtering, is almost completely free of the numerical Cherenkov instability for time steps (scaled to the speed of light) comparable to or smaller than the axial cell size.

3D-PSTD simulation and polarization analysis of a light pulse transmitted through a scattering medium

A tridimensional pseudo-spectral time domain (3D-PSTD) algorithm, that solves the full-wave Maxwell's equations by using Fourier transforms to calculate the spatial derivatives, has been applied to determine the time characteristics of the propagation of electromagnetic waves in inhomogeneous media. Since the 3D simulation gives access to the full-vector components of the electromagnetic fields, it allowed us to analyse the polarization state of the scattered light with respect to the characteristics of the scattering medium and the polarization state of the incident light. We show that, while the incident light is strongly depolarized on the whole, the light that reaches the output face of the scattering medium is much less depolarized. This fact is consistent with our recently reported experimental results, where a rotation of the polarization does not preclude the restoration of an image by phase conjugation.

Scattered-field FDTD and PSTD algorithms with CPML absorbing boundary conditions for light scattering by aerosols

As fundamental parameters for polarized-radiative-transfer calculations, the single-scattering phase matrix of irregularly shaped aerosol particles must be accurately modeled. In this study, a scattered-field finite-difference time-domain (FDTD) model and a scattered-field pseudo-spectral time-domain (PSTD) model are developed for light scattering by arbitrarily shaped dielectric aerosols. The convolutional perfectly matched layer (CPML) absorbing boundary condition (ABC) is used to truncate the computational domain. It is found that the PSTD method is generally more accurate than the FDTD in calculation of the single-scattering properties given similar spatial cell sizes. Since the PSTD can use a coarser grid for large particles, it can lower the memory requirement in the calculation. However, the Fourier transformations in the PSTD need significantly more CPU time than simple subtractions in the FDTD, and the fast Fourier transform requires a power of 2 elements in calculations, thus using the PSTD could not significantly reduce the CPU time required in the numerical modeling. Furthermore, because the scattered-field FDTD/PSTD equations include incident-wave source terms, the FDTD/PSTD model allows for the inclusion of an arbitrarily incident wave source, including a plane parallel wave or a Gaussian beam like those emitted by lasers usually used in laboratory particle characterizations, etc. The scattered-field FDTD and PSTD light-scattering models can be used to calculate single-scattering properties of arbitrarily shaped aerosol particles over broad size and wavelength ranges.

Combined FVTD/PSTD Schemes with Enhanced Spectral Accuracy for the Design of Large-Scale EMC Applications

A generalized conformal time-domain method with adjustable spectral accuracy is introduced in this paper for the consistent analysis of large-scale electromagnetic compatibility problems. The novel 3-D hybrid schemes blend a stencil-optimized finite-volume time-domain and a multimodal Fourier-Chebyshev pseudo-spectral time-domain algorithm that split the overall space into smaller and flexible areas. A key asset is that both techniques are updated independently and interconnected by robust boundary conditions. Also, combining a family of spatial derivative approximators with controllable precision in general curvilinear coordinates, the proposed method launches a conformal field flux formulation to derive electromagnetic quantities in regions with fine details. For advanced grid reliability at dissimilar media interfaces, dispersion-reduced adaptive operators, which assign the proper weights to each spatial increment, are developed. So, the resulting discretization yields highly rigorous and computationally affordable simulations, devoid of lattice errors. Numerical results, addressing detailed comparisons of various realistic applications with reference or measurement data verify our methodology and reveal its significant applicability.

Semi-empirical boundary conditions for the linearized acoustic Euler equations using Pseudo-Spectral Time-Domain methods

Pseudo-Spectral Time-Domain algorithms have emerged as new numerical methods for solving Eulerian problems. These methods, in contrast to more common finite-difference, time-domain approaches, provide isotropic dispersion characteristics. However, the technical literature concerning to this topic presents a serious lack of methods for dealing with partially reflecting boundary conditions in order to simulate surfaces of a specified impedance. In the current paper we present a novel semi-empirical formulation for simulating constant impedance boundary conditions within Pseudo-Spectral techniques based on the Fourier transform. Finally, the validations in one and two dimensions by means of different numerical experiments, show the accuracy of the model.