Space-time spectral methods for parabolic equations: Error analysis of explicit-implicit iteration scheme

The parabolic equation (PE) method is widely used in radiowave propagation predictions. It has the advantages of high efficiency and stability, but it will lead to greater predicting errors in some situations, because the effects of transverse terrain gradients are not modeled. This problem can be solved by extending the 2D PE to the three-dimensional (3D) PE. However, the computing efficiency will degrade because of large scale matrix operations. In this paper, a new method is presented, in which the 3D PE is decomposed into two 2D PEs. It increases the computational efficiency and accuracy effectively. To verify the capability of the proposed method in radiowave propagation prediction, an experiment platform was set up. The computational results using this new method are compared with the experimental and Method of Moment (MoM) numerical computational results. Good agreements are achieved in the comparison. The study of radiowave propagation under complex environments is vitally important for improving the effective signal coverage in wireless communication and television broadcasting with plenty of urban buildings as well as mountainous terrains. It presents a big challenge due to the multiple wave propagation mechanisms involving the line of sight, diffraction and scattering propagation and the characteristic of large scale computation. The major issues refer to how to keep the computational accuracy and conducting the large scale computation at the same time. To meet the request for large scale computation of actual engineering problems, the modified empirical formulas are often used for analysis and computation with the drawback of poor computational accuracy, and the variance of the predicted propagation loss is generally 2 to 5 dB for the flat terrain, but for the complex terrain, the variance of the predicted propagation loss is usually about 5 to 10 dB. Many computational methods have been developed to improve the predicted accuracy. The commonly used methods are geometric diffraction methods, path integration methods and FDTD. It is hard to meet the requirement of engineering computation because these methods are computationally expensive and require high data accuracy of the terrain database at the same time (1). The Parabolic Equation (PE) method is a computational method for radiowave propagation loss besides the full wave method and ray tracing method, and has become a research focus. The PE method has high potential in radiowave propagation of mobile communication and broadcasting and television signal coverage as it can not only deal with complex terrain boundary conditions accurately, but also deal with complex atmosphere waveguide structure. Compared with the full wave method, the PE method deals with an unidirectional wave propagation, in which the equation can be solved by the step marching method in the propagation direction. The PE method can reduce the computational complexity of the full wave equation.

The propagation phenomena through 2D surface duct-to-elevated duct transition is described in terms of parabolic equation (PE) method. The split step marching solution of the PE method is first compared with the reference solutions obtained from Helmholtz wave equation in longitudinally invariant surface duct problem and the excellent agreement between the results is presented. Then the PE method is applied to propagation through surface duct-to-elevated duct transition problem which is non-separable in any of the known orthogonal coordinate system. The transformation of the surface trapped modes into propagating elevated beams is shown with the PE method. >

The parabolic equation (PE) method is very effective for computing the complex electromagnetic environment (EME) simulation. For the large-scale 3D EME simulation based on the PE method, the computation and time complexity is not acceptable. To solve this problem, a quasi-3D PE simulation method with low complexity is introduced. But the accuracy of this method is not high, and some methods have been proposed to improve the accuracy by compensating error, choosing appropriate slice angle, interpolating some values between slices and mixing the PE and ray tracing method. With a multi-level parallel method, the 3D EME simulation can be paralleled from the three levels: job parallel, task parallel, thread parallel, and it is used to accelerate the computation process dramatically in order to calculate faster.

The parabolic equation (PE) method is a useful approach for solving range-dependent ocean acoustics problems. The PE method can also be used to model the propagation of internal gravity waves in the ocean. One of the possible applications of a range-dependent gravity wave model to ocean acoustics would be to generate a model-based noise spectrum that could be used as an alternative or complement to the empirical spectrum of Garrett and Munk [J. Geophys. Res. 80, 291–297 (1975)]. This could be done using the approach of Kuperman and Ingenito for modeling surface-generated acoustic noise [J. Acoust. Soc. Am. 67, 1988–1996 (1980)] for an appropriate distribution of gravity wave sources. The mathematical details of a PE for gravity waves will be discussed. Examples will be presented to illustrate the propagation of gravity waves in range-dependent environments. [Work supported by ONR.]

The parabolic equation (PE) method has been the most efficient range-dependent propagation model in ocean acoustics ever since the split-step Fourier solution was developed [F. D. Tappert, ‘‘Numerical solutions of a canonical nonlinear dispersive wave equation,’’ SIAM Rev. 16, 140 (1974)]. Since the asymptotic accuracy of the split-step Fourier solution is restricted to leading order (a serious limitation for some problems), the PE method was extended to higher-order asymptotic accuracy using Padé approximations. Until recently, the higher-order PE has been solved using relatively inefficient finite-difference solutions, and the split-step Fourier solution has remained in widespread use despite the accuracy improvements. To improve efficiency without sacrificing accuracy, we apply Padé approximations to achieve higher-order accuracy in both the numerics and the asymptotics. The split-step Padé solution, which has been implemented for both fluid and elastic media, achieves both higher-order asymptotic accuracy and the efficiency of the split-step Fourier solution. This finite-difference solution is one to three orders of magnitude faster than other finite-difference solutions.

The ionizable metal vapor released in the ionosphere can artificially generate space electron clouds. They can scatter radio waves at decameter and meter wavelengths so as to realize over-the-horizon propagation. Based on the parabolic equation (PE) method and using Padé-(1,1) polynomial, the electromagnetic scattering characteristics of cylindrical artificial electron clouds are preliminarily studied in this letter. The bistatic radar scattering cross section (RCS) and the ground received field distribution via artificial electron cloud are calculated for various incidence angles at different radio frequencies and distances. The PE method is used here since its advantages of high precision, fast computing and less computing resource occupation over traditional methods. The study of electromagnetic scattering is valuable to the follow-up release experiments for generating electron cloud. It throws a new light on the practical utilization of over-the-horizon radio communication via scattering channel of artificial electron clouds.

An improved fractal sea surface model, which can describe the capillary waves very well, is introduced to simulate the one-dimension rough sea surface. In this model, the propagation of electromagnetic waves (EWs) is computed by the parabolic equation (PE) method using the finite-difference (FD) algorithm. The numerical simulation results of the introduced model are compared with those of the Miller-Brown model and the Elfouhaily spectrum inversion model. It has been shown that the effects of the fine structure of the sea surface on the EWs propagation in the introduced model are more apparent than those in the other two models.

The alternating group explicit (AGE) scheme, together with the alternating direction implicit (ADI) scheme, is used for the parabolic equation (PE) method to analyze the electromagnetic scattering from electrically large conducting targets. The calculation can be taken plane by plane by using the finite difference scheme along the paraxial direction. Furthermore, each transverse plane is divided into two regions, which are a small region around the target and an outer region combined with the air layer and a perfect matching layer. The ADI scheme is applied to the small region, whereas the AGE scheme is introduced to the outer region. In this manner, only a few matrix equations are needed to be solved for the small region and the unknowns of the outer region can be directly obtained without solving any matrix equations. Therefore, the CPU time can be reduced significantly by the proposed method and it can be easily adapted to the parallel technology. The numerical results verify the accuracy and efficiency of the proposed method compared with the Crank-Nicolson-based PE method and the ADI-based PE method. Moreover, the parallel efficiency is given as the last numerical example.

A special-purpose microcomputer-based system with attached array processor has been developed for generating numerical solutions to the parabolic wave equation for on-board applications. We call this system PESOGEN (Parabolic Equation Solution Generator). We discuss the principles of geo-acoustic modeling as implemented on PESOGEN including: water-sediment interface with range-dependent bathymetry, range-dependent gradients in sound speed and loss in sediment layer, sediment-basement interface and artificial absorption, and especially modeling of rough-bottom scattering. We also discuss the principles of pulse response function modeling as implemented on PESOGEN including: band-limited signals and analytic signals, time-domain transmission loss and phase plots, and time-domain reciprocity. Results and examples include propagation studies and computation of pulse response functions in acoustically complex geographical areas. These studies concentrate on the related issues of sensitivity to bathymetry, and rough-bottom scattering and broad-band coherence.

A new theoretical-numerical method has been developed which enables one to accurately and efficiently compute the entire two-dimensional (depth and range) low-frequency underwater acoustic field in realistic ocean environments. The theoretical development is based on the parabolic equation approximation [V.A. Fock, Electromagnetic Diffracfion Problems (Pergamon, 1965)] which automatically includes diffraction and all other full-wave effects, as well as depth- and range-dependent sound speeds and volume losses, and variable bathymetry. This parabolic wave equation is solved numerically using the split-step Fourier algorithm [R. H. Hardin and F. D. Tappert, SIAM Rev. 15, 423 (1973)] which is accurate, efficient, unconditionally stable, and is readily implemented. Operational computer programs based on this method have been used to reliably predict long-range underwater propagation effects in areas of practical interest. Representative results are displayed in the form of detailed computer-drawn decibel contours which illustrates SOFAR, BSR, surface duct, bottom bounce, and other modes of propagation in the presence of horizontal thermoclines, sea-mounts, and sea-ridges, migrating sound channel axes, multiple channels, and so forth.

While GPS is a relatively mature technology, its susceptibility to radio frequency interference (RFI) is substantial. Various investigations, including the Volpe Report (Volpe, 2001) which was the result of a US Presidential Decision Directive (PDD-63) assigned to the Department of Transportation (DOT), have recommended that methods should be developed to monitor, report and locate interference sources for applications where loss of GPS is not tolerable. With GPS becoming an integral utility for developed society, the significance of research projects that enhance and expand the capabilities of GPS RFI localisation is highly important. In response to this requirement for GPS interference localisation, a novel technique called “Inverse Diffraction Parabolic Equation Localisation System” (IDPELS) has been developed. This technique exploits detailed knowledge of the local terrain and an inverse diffraction propagation model based on the Parabolic Equation method (PEM). In wave-propagation theory, an inverse problem may involve the determination of characteristics concerning the source, from field values measured at a certain point or certain regions in space. PEM is an electromagnetic propagation modelling tool that has been extensively used for many applications. This paper will present simulation and field trial results of IDPELS. Simulation results show that this technique has good promise to be useful in locating GPS jamming sources in highly-complex environments, based on networks of GPS sensing antennas. Results also show that the method is capable of locating multiple interference sources. Trials concerning the practical application of IDPELS are also provided. With measured lateral field profiles recorded with a single moving sensor platform in a van, results indicate IDPELS to be a pragmatic localisation technique.

This work provides an introduction to one of the most widely used advanced methods for wave propagation modeling, the Parabolic Equation (PE) method, with emphasis on its application to tropospheric radio propagation in coastal and maritime regions. The assumptions of the derivation, the advantages and drawbacks of the PE, the numerical methods for solving it, and the boundary and initial conditions for its application to the tropospheric propagation problem are briefly discussed. More details are given for the split-step Fourier-transform (SSF) solution of the PE. The environmental input to the PE, the methods for tropospheric refractivity profiling, their accuracy, limitations, and the average refractivity modeling are also summarized. The reported results illustrate the application of finite element (FE) based and SSF-based solutions of the PE for one of the most difficult to treat propagation mechanisms, yet of great significance for the performance of radars and communications links working in coastal and maritime zones — the tropospheric ducting mechanism. Recent achievements, some unresolved issues and ongoing developments related to further improvements of the PE method application to the propagation channel modeling in sea environment are highlighted.

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

The parabolic equation method provides an excellent combination of accuracy and efficiency for range-dependent ocean acoustics and seismology problems. This approach is highly developed for problems in which the ocean bottom can be modeled as a fluid. For the elastic case, there remain some accuracy limitations for problems involving sloping interfaces. Progress on this problem has been made by combining a new formulation of the elastic parabolic equation that handles layering more effectively [W. Jerzak, ‘‘Parabolic Equations for Layered Elastic Media,’’ doctoral dissertation, Rensselaer Polytechnic Institute, Troy, NY (2001)] and a mapping approach that handles sloping interfaces accurately [J. Acoust. Soc. Am. 107, 1937–1942 (2000)]. This approach makes it possible to handle problems involving complex layering and steep slopes, but the rate of change of the slope must be small. The method and its application to data will be described. Our immediate goal is to model propagation of seismic surface waves propagating across a transition between dry and marshy terrain. We have suitable data applicable to vehicle-tracking problems from Marine Corps Base Camp, Pendleton, CA. [Work supported by ONR.]