Parabolic Equation Method Research Articles

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.

A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations

An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.

Enhancing Communication Link Predictions for Ingenuity Mars Helicopter Mission With the Parabolic Equation Method

Enhancing Communication Link Predictions for Ingenuity Mars Helicopter Mission With the Parabolic Equation Method

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Abstract This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

Optimal convergence analysis of weak Galerkin finite element methods for parabolic equations with lower regularity

Electromagnetic scattering of cylindrical artificial electron cloud by using parabolic equation method

Abstract 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.

Underwater Acoustic Propagation using Monterey-Miami Parabolic Equation in Shallow Water Kayeli Bay Buru Distric

Indonesia's geographical position is an advantage compared to other countries, both in terms of geoeconomics, geopolitics and geostrategy. For this reason, it is necessary to develop and use acoustic methods to describe underwater features, carry out underwater communications or to measure oceanographic variables at sea. This research was intended to provide an analytical and visual graphical description with the aim that it can be used for various purposes both in the research, military and other marine fields, as well as to analyze the influence of sediment and different frequencies on acoustic propagation patterns in shallow waters of Kayeli Bay. This research was conducted using CTD data from Kayeli Bay, which is a body of water in Buru Regency, Maluku Province and is located between 3° 15' 55'' – 3° 22' 50" S and 127° 01'35" – 127° 01' 35 "E, using the Monterey-Miami parabolic equation method using 4 types of sediment and 3 different frequencies as model input. From the results of this research it can be concluded that the propagation of sound waves in shallow seas is greatly influenced by the type of sediment and frequenty used. Changes in acoustic impedance at the bottom of the water and within the water column can significantly influence the behavior of acoustic waves in shallow water environments, and accurate acoustic impedance data are critical for effective ray tracing modelling.

A modified domain decomposition spectral collocation method for parabolic partial differential equations

<p>In this paper, utilizing Legendre polynomials as the basis functions in both space and time, we present a modified domain decomposition spectral method for 2-dimensional parabolic partial differential equations. For solving the obtained linear/nonlinear algebraic equations, a dimension expanding preconditioner is applied employing the obtained saddle construction of the coefficient matrix. Numerical examples are given to show the performance of the presented method and the efficiency of the preconditioner.</p>

Analysis of a fourth-order compact $ \theta $-method for delay parabolic equations

<abstract><p>The upper bounds for the powers of the iteration matrix derived via a numerical method are intimately related to the stability analysis of numerical processes. In this paper, we establish upper bounds for the norm of the <italic>n</italic>th power of the iteration matrix derived via a fourth-order compact $ \theta $-method to obtain the numerical solutions of delay parabolic equations, and thus present conclusions about the stability properties. We prove that, under certain conditions, the numerical process behaves in a stable manner within its stability region. Finally, we illustrate the theoretical results through the use of several numerical experiments.</p></abstract>

Comparative analysis of finite‐difference and split‐step based parabolic equation methods for tunnel propagation modelling

AbstractRadio wave propagation modelling in railway environments is of fundamental importance in designing reliable train communication systems. Parabolic equation (PE) methods have been widely applied to the modelling of wave propagation in tunnels due to their high computational efficiency and fidelity. The finite‐difference parabolic equation (FDPE) and the split‐step parabolic equation (SSPE) methods are two commonly used approaches to solve PE numerically. However, the relevant literature is still missing a comprehensive study of their performance, including the selection of parameters such as discretisation steps and the tradeoffs involved in terms of their accuracy and efficiency, especially as current wireless systems shift to high frequencies. In this study, a systematic analysis of the error and computational complexity of the FDPE and SSPE methods for radio wave propagation modelling in tunnels is provided. Guidelines for the choice of their parameters are provided, and their performance is demonstrated through both numerical examples and experimental measurements in actual tunnel cases.

Validation of estimated underwater radiated noise propagation from a ship using the parabolic equation method by underwater sound measurements in actual sea

In 2022, the International Maritime Organization started a review of the nonmandatory guidelines for the reduction of underwater radiated noise (URN) from ships and a discussion about the next step. In the case of considering the extent of underwater radiated noise from a ship, the propagation of the URN and the background noise in the sea area should be considered in addition to the source level of the URN. In this study, the time series of the received levels of underwater sound measured off Izu-Oshima Island, Japan, are shown in the 1/3 octave band level. The time series of the received levels were compared with the propagation loss estimated by the parabolic equation method to validate the accuracy of the estimation of the propagation of underwater sound in the measurement area.

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

Studying the influence of cold-core mesoscale ocean eddies on sound propagation based on the parabolic equation method

Mesoscale vorticity is an important factor that affects the non-uniform horizontal and vertical distribution of hydrologic elements in the ocean. In this paper, temperature and salinity data from measured mesoscale cold-core eddies during a voyage within a certain sea area are studied, and a synchronous sound field test was conducted on the mesoscale vorticity. Based on the temperature and salinity data, the parabolic equation (PE) method was selected to predict the acoustic field, and the results were compared with the measured propagation loss data of the acoustic field to verify the accuracy and effectiveness of the PE model. The underwater sound propagation characteristics in the mesoscale cold-core eddy environment were then analyzed using the temperature and salinity data retrieved from the voyage. It was found that the convergence area of the acoustic field gradually dispersed with an increase in the depth of the sound source. In a mesoscale eddy environment, when sound waves propagate from inside to outside the eddy, the presence of a cold-core eddy causes the convergence area to shift toward the edge of the eddy, and the deviation gradually decreases with an increase in the depth of the sound source.

Development of the parabolic equation for calculation of sea waves diffraction in port area

Application method of parabolic equations is presented in this paper for calculating diffraction of sea waves behind converging breakwaters, which entrance is not parallel to front of approaching waves. For this, the method of linear superposition of results obtained separately for each breakwater was used. Based on a comparison of results obtained by this method with results of physical (obtained in a wave basin) and numerical (obtained using DHI MIKE 21 BW) model experiments with different settings (40 models in total), a conclusion about using allowability of the obtained equations was made. Results of the study make it possible to recommend the obtained equations for practical use in studies of seaports wave regimes, where diffraction phenomena are strong. Complexity of a function used in the parabolic method causes appearance of “petals” of diffraction coefficient isolines in protected water area. Approximate equations are presented for smoothing the oscillations of the complex amplitude function along lines parallel and perpendicular to axis of breakwaters. It is shown that associated error in obtaining diffraction coefficient varies on average within 2–5 %, and maximum error obtained was 12.5 %.

Generalisable convolutional neural network model for radio wave propagation in tunnels

AbstractPropagation models are essential for the prediction of received signal strength and the planning of wireless systems in a given environment. The vector parabolic equation (VPE) method has been widely applied to the modelling of radio wave propagation in tunnels. However, carrying out simulations for large‐scale environments is still computationally expensive. A convolutional neural network (CNN)‐based propagation model, which can provide high‐fidelity received signal strength prediction based on results from low‐cost VPE simulations, is proposed. A thorough study of the generalisability, including both interpolation and extrapolation capabilities, of the proposed CNN model is conducted. It is found that the proposed model can achieve significant computational savings while maintaining acceptable accuracy, and its performance is validated in both simulations and actual tunnel cases.

Shadow zone mapping using parabolic equation method in the Bali Strait Waters

Shadow Zone is an area where sound waves cannot penetrate a water column due to deflection. This study aims to map the shadow zone based on oceanographic parameters in the Bali Strait. The data used are temperature, salinity, and depth for 31 days in the East Season (July) in the Bali Strait at two stations from the results of numerical modeling downloaded from the website https://marine.copernicus.eu/. Shadow zone simulation uses the parabolic equation method with certain parameter limits. The vertical temperature profile shows that the deeper the water is, the lower the temperature, whereas salinity is the opposite. The sound velocity is affected by the parameters of temperature and salinity. At the location of the 25 m transducers, the undetected medium beam is narrower than the location of the 110 m transducers so that the shadow zone with a 110 m transducer position has a broader area. The Shadow zone area in the northern Bali Strait is effectively detected using a frequency of 100 Hz. In comparison, in the southern part of the Bali Strait, it is effectively seen at a frequency of 38 Hz, with the respective transducer positions used at a depth of 25 and 110 m.

Wind turbine sound propagation: Comparison of a linearized Euler equations model with parabolic equation methods.

Noise generated by wind turbines is significantly impacted by its propagation in the atmosphere. Hence, for annoyance issues, an accurate prediction of sound propagation is critical to determine noise levels around wind turbines. This study presents a method to predict wind turbine sound propagation based on linearized Euler equations. We compare this approach to the parabolic equation method, which is widely used since it captures the influence of atmospheric refraction, ground reflection, and sound scattering at a low computational cost. Using the linearized Euler equations is more computationally demanding but can reproduce more physical effects as fewer assumptions are made. An additional benefit of the linearized Euler equations is that they provide a time-domain solution. To compare both approaches, we simulate sound propagation in two distinct scenarios. In the first scenario, a wind turbine is situated on flat terrain; in the second, a turbine is situated on a hilltop. The results show that both methods provide similar noise predictions in the two scenarios. We find that while some differences in the propagation results are observed in the second case, the final predictions for a broadband extended source are similar between the two methods.

Analysis of Radar Target Scattering Echo With Surface Ducting in Large-Scale Environments Based on the PE-MoM Hybrid Method

This letter proposes a three-dimensional (3-D) PE- MoM hybrid method to predict radar target scattering with sur- face ducting in large-scale environments. The method divides the complex multiscale problem into two sub-problems: electromag- netic (EM) wave propagation and scattering. The bidirectional 3-D parabolic equation (PE) method is used to consider the influence of environmental factors such as irregular terrain and atmospheric duct on EM wave propagation, while the method of moments (MoM) is used to calculate the scattering from the key target. By introducing phase compensation and Fourier synthesis techniques, the proposed method can be extended to predict transient scattering echoes from the target. A typical example is given to verify the correctness of the proposed method, and good consistency is also seen from comparison with the MoM. Finally, simulation analysis is carried out for the scattering of a missile target in a sea-land composite environment. The results show that the proposed method can effectively predict the scattering echoes of low-altitude targets in large-scale environments.

Unconditionally convergent and superconvergent finite element method for nonlinear time-fractional parabolic equations with distributed delay

Unconditionally convergent and superconvergent finite element method for nonlinear time-fractional parabolic equations with distributed delay

A modified weak Galerkin finite element method for parabolic equations on anisotropic meshes

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method.