Parabolic Equation Method Research Articles

Theoretical and experimental study of diffraction by a thin cone

A problem of diffraction of ultrasound acoustic waves by an acute-angled hard cone is studied. Within the framework of the parabolic equation method, an analytical solution of the problem for an arbitrarily located point source is built. Specifically, the problem is reduced to the Volterra boundary integral equation, which can be solved using the Fourier transform. An experimental measurement of the diffracted field is carried out. The experiment is based on the MLS method adapted for narrowband sound sources. A comparison of experimental and theoretical results is provided.

Discontinuous galerkin method of the first order parabolic differential equation

The objective of this paper is to propose a modest numerical error analysis by applying DG finite element method for the parabolic differential equation. The DG method is an imperative numerical method with much mass compensation and more flexible meshing than other numerical methods. The DG method starts by discretizing the domain into a set of non-overlapping elements. This study gives a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Investigation of microwave power transfer near sea surface with 2D parabolic equation method

AbstractDue to the existence of the evaporation duct near‐sea‐surface described as an inhomogeneous refractive index of atmosphere, microwaves can be trapped in the duct and propagate along it. To investigate wireless power transfer (WPT) over the sea surface, we utilize the two‐dimensional parabolic equation to calculate the electric field distribution and to obtain the transferred power. By analysing the propagation results with specific evaluation functions, we obtained the proper conditions for long‐distance WPT over the sea surface. When the initial field distributes uniformly on the one‐meter aperture, the frequency is 12 GHz, and the centre position is 4 m, a unique focusing spot can be obtained in the duct and nearly 5%, 8%, and 12% of the initial power remains at a distance of 10 km with a receiving aperture of 1, 2, and 3 m, respectively. The presented WPT scheme near the sea surface has potential applications in the power supply or charging for ships or small islands.

Deep learning methods for modeling infrasound transmission loss in the middle atmosphere

Accurate modeling of infrasound transmission losses (TLs) is essential to assess the performance of the global International Monitoring System (IMS) infrasound network. Among existing propagation modeling tools, parabolic equation method (PE) enables TLs to be finely modeled, but its computational cost does not allow exploration of a large parameter space for operational monitoring applications. To reduce computation times, Brissaud et al. (2022) explored the potential of convolutional neural networks (CNNs) trained on a large set of regionally simulated wavefields (>1000 km from the source) to predict TLs with negligible computation times compared to PE simulations. However, this new method shows difficulties in upwind conditions, especially at low frequencies, and causal issues with winds at large distances from the source affecting ground TLs close to the source. In this study, we have developed an optimized CNN network designed to minimize prediction errors while predicting TLs from globally simulated combined temperature and wind fields spanning over propagation ranges of 4000 km. Our approach enhances the previously proposed one by implementing key optimizations that improve the overall architecture performances. The implemented model predicts TLs with an average error of 20 dB in the whole frequency band (0.1-4 Hz) and explored realistic atmospheric scenarios.

Eolization: A project on spatial auralisation of wind turbine noise

The increase of onshore wind farm projects in Europe has heightened concerns regarding the noise emitted by wind turbines and the potential nuisance to nearby residents. To address this issue, advanced tools are required for accurately assessing and predicting the noise levels and perceived annoyance from wind turbines. Physics-based auralization emerges as a promising approach for recreating with fidelity wind turbine noise in controlled environments. To enhance the immersive experience, this study proposes to extend the current methodologies with the spatial representation of wind turbine noise using ambisonics. In this work, the sound emitted by each turbine is computed using an extended source model. The propagation is simulated with a parabolic equation method accounting for the flow surrounding the wind turbines. Finally, the signal is auralized and spatialized using spherical harmonic decomposition. This methodology provides a complete framework for predicting and reproducing the three-dimensional time-varying sound field around the listener. Overall, these improvements offer valuable insights into the spatial characteristics of wind turbine noise.

Application of Slip Interface Conditions in the Elastic Parabolic Equation

The parabolic equation method is the most effective approach for solving wave propagation problems in environments with strong vertical variations and gradual horizontal variations. This approach efficiently provides accurate solutions for problems involving sloping fluid-fluid interfaces, sloping solid-solid interfaces, and sloping solid boundaries. There remains a need for improvement for the case of sloping fluid-solid interfaces. The most promising approach to date is the combination of a single-scattering approximation and the treatment of part of the fluid layer as a solid layer with low shear speed. This is a singular perturbation, which results in rapid oscillations near the interface between the low-speed layer and the true solid layer. It is demonstrated here that the rapid oscillations may be eliminated by using slip conditions at the interface between the low-speed layer and the true solid layer. Stability problems were encountered when attempting to directly implement the slip conditions, but this problem was resolved by using an equivalent condition involving the normal derivative of the tangential displacement. This progress does not fully resolve the case of the sloping fluid-solid interface. There remains a need to find a vertical interface condition that is compatible with the slip conditions and that provides accurate results for problems involving sloping fluid-solid interfaces.

A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

A New Reduced Basis Method for Parabolic Equations Based on Single-Eigenvalue Acceleration

Acoustic Propagation in the Near‐Surface Martian Atmosphere

AbstractThis work introduces a comprehensive model of sound propagation on Mars, in light of the recent operation of several microphones on the Martian surface. The main outcome of this work is an operational acoustic model capable of simulating the sound field created by any source, at any location on the Martian surface, at any time. Expanding on the result of previous work (Gillier et al., 2024, https://doi.org/10.1029/2023je008257), we use the parabolic equation method for sound propagation in order to obtain the overall sound field produced by a source, in a given atmospheric composition and state, and accounting for ground properties. The resulting model enables the study of acoustics on Mars, and has the potential also to be used to probe the properties of the Martian environment using acoustic measurements with known sources. We investigate the effects of the Martian ground and the vertical profile of temperature and wind, on sound propagation. We find that the ground has a minor effect on sound propagation, and the wind profile strongly influences sound propagation as on Earth. However, the midday near surface temperature profiles on Mars are shown to cause refraction, which generates non‐negligible acoustic losses that are an order of magnitude stronger than typical refraction‐related acoustic losses on Earth. We show that the effect of the Martian atmospheric turbulence is to slightly reduce the acoustic losses due to refraction. Finally, we apply our model to show that refraction and atmospheric turbulence have a negligible effect on the propagation of sound from Ingenuity to the Perseverance rover.

Galerkin-finite difference method for fractional parabolic partial differential equations

The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.

Three-dimensional acoustic propagation of noise from impact pile driving in a complex costal environment and its effects on large yellow croaker (Pseudosciaena crocea)

The effects of high-intensity impulsive noise generated by pile driving on fish are a major concern in environmental impact assessments. Numerical acoustic models are essential for predicting underwater-acoustic-related problems in complex coastal environments prior to offshore construction. However, underwater noise modeling for impact pile driving has often been performed using simplistic propagation models that are inadequate for three-dimensional (3D) environments. A 3D parabolic equation method (PE) was established in this study to better predict broadband transmission loss (TL) from impact pile driving in complex coastal environments and its influence on the large yellow croaker (Pseudosciaena crocea). The effects of 3D propagation were investigated using two realistic scenarios with different bathymetric complexities. The values and attenuation rate of the broadband TL for the steeply sloped bottom were significantly greater than those for the flat and weakly varying bottoms over 3 km. At a water depth of 5 m, a difference of approximately 10 dB was observed between the two TL scenarios at a distance of 4.5 to 5 km. The simulation results are in reasonable agreement with the field measurement data, with a difference of less than 3 dB. The zones of behavioral response and injury in the large yellow croaker were estimated using the For3D model. The results showed that the effects of the noise generated by the impact pile driving on the large yellow croaker were evident and three-dimensional. Therefore, 3D propagation effects should be considered when analyzing the influence of underwater noise on marine animals.

An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Discontinuous Galerkin method for linear parabolic equations with L^1-data

Discontinuous Galerkin method for linear parabolic equations with L^1-data

Moving planes and sliding methods for fractional elliptic and parabolic equations

Abstract In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.

Discontinuous Galerkin Method of the First Order Parabolic Differential Equation

This paper is concerned with the mathematical study of the discontinuous Galerkin (DG) finite element method for the parabolic differential equation. The DG method is a vital numerical method with much mass compensation and more flexible meshing than other methods. In this study, we give a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The error analysis of this problem is also established. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Mapping past and future shipping noise in European seas

Against the backdrop of a steadily increasing demand for sea transport of goods and people, the development of a reliable marine shipping soundscape model is an essential planning requirement to assess the effect on ocean noise of operational and technological changes aimed at mitigating the environmental impact of the shipping sector. The NAVISON (Navis Sonus) project, conducted with the support of the European Maritime Safety Agency, employs a specially developed parametric vessel source model with the objective of producing shipping sound maps in European seas for past, present, and potential future conditions over a time span from 2016 to 2050. The source model is combined with historical ship tracking data from the automated identification system (AIS), or projected shipping densities and mitigation scenarios, to calculate spatial ship noise emissions data for input to a sound mapping tool. The mapping tool computes underwater sound propagation using the parabolic-equation method, drawing upon ocean-scale databases of bathymetric, oceanographic, and sediment properties. Project outputs are provided as map layers of sound pressure level and sound energy according to vessel type, season, region, year, and operational conditions; from these layers, maps can be generated for user-specified combinations of mitigation measures. Maps are presented in two frequency bands (centred at 63 Hz and 125 Hz) selected for assessing Good Environmental Status in the context of the European Union’s Marine Strategy Framework Directive.

The statistical characteristics of the atmospheric anisotropic inhomogeneities and their effect on the intensity fluctuations of infrasound field

The influence of anisotropic wind velocity and temperature inhomogeneities on the attenuation of infrasound field intensity with increasing distance from a point source and on its altitude distribution is studied. The field is calculated as a function of receiver height and horizontal distance from the source using method of the pseudo-differential parabolic equation for the atmosphere with model realizations of anisotropic effective sound speed fluctuations. These realizations are obtained from the nonlinear shaping model for the gravity wave perturbations which produces the fluctuations with both the vertical and horizontal spectra consistent with the observed spectra. When propagating in the stratospheric and thermospheric wave guides the multiple scattering of infrasound field from the anisotropic fluctuations results in certain vertical wave number spectra and probability density functions of infrasound intensity fluctuations in the stratospheric (altitudes 30–40 km) and mesospheric layers (50–70 km). The statistical characteristics of the intensity fluctuations as a function of distance from the source (up to 2200 km) were studied. The same characteristics were obtained for the infrasound field scattered from the inhomogeneities whose vertical profile was retrieved from the infrasound signals from surface explosions detected in the shadow zone.

The variable two-step BDF method for parabolic equations

The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios rj:=kj/kj-1⩽1.8685\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_j:=k_j/k_{j-1}\\leqslant 1.8685$$\\end{document} and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398.