Parabolic Equation Approximation Research Articles

Error estimates for the semidiscrete finite element approximation of linear nonlocal parabolic equations

Existence and uniqueness are proved for nonlocal (in time) for solutions of linear parabolic partial differential equations. Instead of an initial condition, there is a relation connecting the initial value to values of the solution at other times. L2 error estimates are obtained for the semidiscrete approximation of the problem using finite elements in the space variables.

On whispering gallery wave propagation in a medium with a vertical separating boundary

The solution of the problem of transmission of a whispering gallery wave through the separating boundary is constructed in the parabolic equation approximation. Formulas for the transformation coefficients of the normal modes are constructed in the case of the Dirichlet and Neumann problems. Computational results are presented.

Time domain analysis of normal mode, parabolic, and ray solutions of the wave equation

Estimation of sound speed and temperature fields from tomographic data requires knowledge of the accuracy of the forward model. For this reason, time domain results using three models derived from the acoustic wave equation are compared here. The models chosen for the analysis are normal modes (NM) and the parabolic equation (PE) and ray approximations. (The NM and PE time series are obtained via Fourier synthesis). The analysis is done using a depth-dependent sound-speed profile where only refracted and surface-reflected/refracted energy is allowed to propagate. Because the NM representation is theoretically exact in this environment, it is used as the comparative base. Travel-time sequences generated using the PE and ray approximations are compared to those generated using NM. For the angular aperature analyzed (±9 to ±16.5 deg), phase errors in the PE models investigated limit reliable predictions of travel time to distances of about 1000 km. However, the ray approximation is found to be adequate for estimating arrival times at ranges out to 4000 km for frequencies of 100 Hz and above.

Matched-field processing in a range-dependent environment

Matched-field processing (MFP) is being considered increasingly for three-dimensional (3-D) localization of an acoustic source in a noisy ocean environment. MFP consists of comparing the measured acoustic field to the full field computed using a geophysical and a propagation model. Most MFP implementations have involved only range-independent propagation models, and many have been restricted to vertical arrays. However, many realistic environments cannot be adequately described by range-independent models, and the problem of localization using more general arrays is of increasing interest. In this paper, a technique is described for range-dependent MFP with arbitrary arrays, where the field is computed using a parabolic equation (PE) approximation. Using PE, two-dimensional (2-D) field values are computed for each sensor in the array for a set of possible source ranges, depths, and (N) bearings to form an N×2-D field model. Discrete estimates of the position of the source are obtained by applying MFP, with this range-dependent model providing field values at the sensors for possible source positions. Using simulated data, source localization in a noisy range-dependent environment is illustrated for several arrays. For the particular cases studied, a lower level of ambiguity was observed for range-dependent MFP than for a related range-independent problem. The use of range-dependent models in MFP should significantly improve array design and analysis of real data from many locations.

Optimal absorber potentials in PE modeling

The parabolic equation (PE) approximation—essential in solving range‐dependent propagation problem—relies on iterated fast Fourier transforms (FFTs). Preventing significant errors due to the periodicity of the finite FFT (“ringing”) requires inserting an absorber potential to confine the wave energy to the range of interest. Such potentials, or sponges, have, to date, been strictly ad hoc. The problem of creating an optimal sponge by using an optical model analogy of the reflectionless potential U = U0* [sech(a*x)]2 of 1‐D quantum mechanics is studied. This approach permits the development of closed‐form expressions for both transmission and reflection coefficients in terms of elementary transcendental and gamma functions of complex argument. Using these coefficients and an efficient technique for evaluating such gamma functions yields a quantitative study of the problem of the optimal sponge. These results demonstrate that, for relevant parameter domains, there may be several “branches” of minimal paths ...

Compensation-theorem analyses of ground wave propagation over irregular, inhomogeneous terrains

Work during the last 40 years on various aspects of ground wave propagation is surveyed, including dipole propagation over flat homogeneous and inhomogeneous terrains, smooth homogeneous earth, and irregular homogeneous and inhomogeneous terrains. The Ott-Volterra integral equation for TM p ground wave field over irregular homogeneous paths is discussed and an occasional troublesome behavior of the modified Sommerfeld attenuation function explained. Several representations of spatially distributed transmitter current are presented. A Compensation-theorem Volterra for the ground wave over inhomogeneous terrain is derived, in the parabolic wave equation approximation. A recent Compensation-theorem integral equation for general terrain satisfying the hyperbolic wave equation is reviewed. Computations for several very rough terrains indicate the ‘parabolic’ integral equations may often be more accurate.

Extended Huygens-Kirchhoff method for predicting transient sound propagation through inhomogeneous moving media

The linear acoustics equations for when the ambient medium has inhomogeneous flow yield an integral corollary analogous to that originally derived by Helmholtz for the reduced wave equation and by Kirchhoff for the wave equation. Such a corollary involves a Green's function for the ambient medium that is here approximated using the eikonal approximation following the technique of Sommerfeld and Runge. The corollary enables one to predict transient waveforms at distant points from information prescribed over a surface and improves upon ordinary geometrical acoustics in that it can account for the presence of caustics in the intervening space. Huygens' principle is realized by this corollary and geometrical acoustics and parabolic equation approximations arise as limiting cases. Numerical examples are given for some simple cases of flows and the use of the overall technique to explore sonic boom propagation in the turbulent atmosphere is outlined. Although the Green's function is approximate, the relation is a good approximate technique for advancing the overall field over moderate distances, without having to explicitly consider the field at intermediate points. [Work supported by NASA-LRC and by the William E. Leonhard endowment to Pennsylvania State University.]

Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity

Semidiscrete finite element methods for a semilinear parabolic equation in R d {R^d} , d ≤ 3 d \leq 3 , were considered by Johnson, Larsson, Thomée, and Wahlbin. With h the discretization parameter, it was proved that, for compatible and bounded initial data in H α {H^\alpha } , the convergence rate is essentially O ( h 2 + α ) O({h^{2 + \alpha }}) for t positive, and for α = 0 \alpha = 0 this was seen to be best possible. Here we shall show that for 0 ≤ α > 2 0 \leq \alpha > 2 the convergence rate is, in fact, essentially O ( h 2 + 2 α ) O({h^{2 + 2\alpha }}) , which is sharp.

Hybrid (ray)-(parabolic equation) analysis of propagation in ocean acoustic guiding environments

Acoustic ray theory provides an effective means for predicting propagation in general environments. Ray theory fails in the vicinity of caustics and must there be augmented by uniform asymptotic treatment. However, even such corrections are inadequate at long ranges in a ducting channel, where the caustics due to multiple reflected and/or refracted rays accumulate at locations near the duct axis. To repair this deficiency of ray acoustics, the caustic forming ray spectra have previously been replaced by guided modes in a self-consistent ray-mode scheme. It is now shown that the troublesome ray spectra at long ranges can be replaced as well by a narrow-angle parabolic equation (PE) approximation to the full pressure field. Taken together, the hybrid ray-PE form can be regarded either as correcting via PE the failure of ray acoustics near the guiding axis or as removing the narrow-angle restriction from PE by filling the wide angle spectra with rays. The theory is developed first for a range-independent surface duct, and then for a range-dependent guiding-to-antiguiding transition. Unlike the adiabatic modes, the numerical PE algorithm can accommodate changes from guiding to antiguiding. The analogous problem of propagation along a circular concave boundary and along a boundary with concave–convex transition is treated briefly in the Appendix.

Some remarks on the Galerkin approximation of parabolic equations

Some remarks on the Galerkin approximation of parabolic equations

A local description of the directional characteristics of the propagation of electromagnetic fields

A method is presented for determining a local measure of the amount of electromagnetic field propagation as a function of direction. This determination is made for the region near a given point by using the known field in the vicinity of that point within a given plane. The method is developed by assuming the parabolic equation approximation. Then the resulting equations are cast in the path integral form. A reformation of the path integral formulation is then developed. This reformation contains a quantity which is interpreted as a reasonable measure of the amount of field (amplitude) propagating in each direction. The relationship of this model to the plane-wave expansion technique is noted. The use of the method in two dimensions is illustrated by several numerical calculations. These numerical calculations are shown to be consistent with the predictions of the geometric theory of diffraction. >

Perturbation Theory for Continuous Stochastic Equations

The various general perturbational schemes for continuous stochastic equations are considered. These schemes have many analogous features with the iterational solution of Schwinger equation for S-matrix. The following problems are discussed: continuous stochastic evolution equations for probaibility distribution functionals, evolution equations for equal time correlators, perturbation theory for Gaussian and Poissonian additive noise, perturbation theory for birth and death processes. stochastic properties of systems with multiplicative noise. The general results are illustrated by diffusion - controlled reactions, fluctuations in closed systems with chemical processes, propagation of waves in random media in parabolic equation approximation, and nonequilibrium phase transitions in systems with Poissonian breeding centers. The rate of irreversible reaction X + X → A (Smoluchowski process) is calculated with the use of general theory based on continuous stochastic equations for birth and death processes. The threshold criterion and range of fluctuational region for synergetic phase transition in system with Poissonian breeding centers are also considered.

Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data

We consider time-continuous spatially discrete approximations by the Galerkin finite element method of initial-boundary value problems for semilinear parabolic equations with nonsmooth or incompatible initial data. We find that the numerical solution enjoys a gain in accuracy at positive time of essentially two orders relative to the initial regularity, as a result of the smoothing property of the parabolic evolution operator. For higher-order elements the restriction to two orders is in contrast to known optimal order results in the linear case.

On Galerkin approximations of parabolic equations in time dependent domains

On Galerkin approximations of parabolic equations in time dependent domains

A mathematical model for the three-dimentional ocean sound propagation

A model is developed mathematically to represent sound propagation in a three-dimensional ocean. The complete development is based on characteristics of the physical environment, mathematical theory, and computational accuracy. While the two-dimentional underwater acoustic wave propagation problem is not yet solved completely for range-dependent environments,three-dimentional environmental effects, such as fronts and eddies, often cannot be neglected. To predict underwater sound propagation, one usually deals with the solution of the Helmholtz (reduced wave) equation. This elliptical equation, along with a set of boundary conditions including a wall condition at the maximum range, forms a well-posed problem, which is pure boundary-value problem. An existing approach to economically solve this three-dimensional range-dependent problem is by means of a two-dimensional parabolic partial differential equation. This parabolic approximation approach, within the limitation of mathematical and acoustical approximations, offers efficient solutions to a class of long-range propagation problems. The parabolic wave equation is much easier to solve than the elliptic equation; one major saving is the removal of the wall boundary condition at the maximum range. The application of the two-dimensional parabolic wave equation to a number of realistic problems has been successful. We discuss the extension of the parabolic equation approach to three-dimensional problems. This paper begins with general considerations of the three-dimensional elliptic wave equation and shows how to transform this equation into parabolic equations which are easier to solve. The development of this paper focuses on wide angle three-dimensional underwater acoustic propagation and accommodates as a special case prevoius developments by other authors. In the course of our development, the physical properties, mathematical validity, and computational accuracy are the primary factors considered. We describe how parabolic wave equations are derived and how wide angle propagation is taken into consideration. Then, a discussion of the limitations and the advantages of the parabolic equation approximation is highlighted. These provide the background for the mathematical formulation of three-dimensional underwater acoustic wave propagation models. Modelling the mathematical solution to three-dimensional underwater acoustic wave propagation involves difficulties both in describing the theoretical acoustics and in performing the large scale computations. We have used the mathematical and physical properties of the problem to simplify considerably. Simplications allow us to introduce a three-dimensional mathematical model for underwater acoustic propagation predictions. Our wide angle three-dimensional parabolic equation model is theoretically justifiable and computationally accurate. This model offers a variety of capabilities to handle a class of long-range propagation problems under acoustical environments with three-dimensional variations.

A discussion on the angle of propagation in relation to the parabolic equation approximation

In approximating the reduced wave equation by parabolic-like partial differential equations, it is interesting to observe that in the direction of propagation, the size of the angle is related to the order of the parabolic-like equation. The order of the Padé rational function approximation used in the derivation of the parabolic equation (PE) determines the order of the parabolic equation. The “standard PE” and the “wide-angle PE” can be thought of as having been derived through approximations by a Padé (1,0) and a Padé (1,1), and can be considered to have orders 2 and 3, respectively. In this paper, we discuss parabolic-like approximations using higher-order Padé rational functions which result in higher-order pseudopartial differential equations. Their behavior in relation to the angle of propagation is examined.

Linear time-invariant space-variant filters and the parabolic equation approximation

Wave propagation in a random, inhomogeneous ocean is treated as transmission through a linear, time-invariant, space-variant, random communication channel. Using the parabolic equation approximation of the Helmholtz wave equation, a random transfer function of the ocean volume is derived. The ocean volume is characterized by a three-dimensional random index of refraction which is decomposed into deterministic and random components. Two additional calculations are performed using the transfer function. The first involves the derivation of the equations for the random, output electrical signals at each element in a receive planar array of complex weighted point sources in terms of the frequency spectrum of the transmitted electrical signal, the transmit and receive arrays, and the transfer function of the ocean medium. The second involves the derivation of the coherence function.

A three-dimensional model for acoustic scattering from rough ocean bathymetry

Scattering of acoustic waves by rough bathymetry is an important factor influencing the performance of array systems. This paper discusses a computer model that calculates the directional distribution of bottom-scattered energy received at an array. The model simultaneously includes the effect of bathymetric scattering and refraction caused by a varying sound-speed structure in the water column. The numerical techniques used include the split-step solution to the parabolic equation approximation to the Helmholtz equation away from the boundary, propagation of the coherence function near the boundary, and a modified Kirchhoff formulation to incorporate scattering. An exponential substitution approximation, which can reduce the computation time of the scattering component by two orders of magnitude with little loss in accuracy, is described.

The state-of-the-art parabolic equation approximation as applied to underwater acoustic propagation with discussions on intensive computations

The Parabolic Equation (PE) has applications in many different scientific fields such as electromagnetics, optics theory, quantum mechanics, plasma physics, seismology, underwater acoustics, etc. This presentation centers on a discussion of the parabolic equation approximation as applied to underwater acoustic wave propagation—past, present, and future directions. A review will be given of past contributions. Recent developments are highlighted. Looking ahead we discuss what the parabolic equation method can do in order to stimulate future research and development, as well as applications. Intensive computations with respect to the parabolic equation implementation will also be discussed.

Convergence Estimates for Semidiscrete Approximations of Nonselfadjoint Parabolic Equations

Semidiscrete Galerkin approximations of the second order nonselfadjoint parabolic equations are considered. $L_2$-error estimates for $L_2$-initial conditions are derived by use of semigroup methods and of analyticity of the underlined semigroup. The scheme here presented can be also freely applied to nonconforming elements.