Wave Equation Method Research Articles

Unconditionally Superconvergence Error Analysis of an Energy-Stable and Linearized Galerkin Finite Element Method for Nonlinear Wave Equations

Unconditionally Superconvergence Error Analysis of an Energy-Stable and Linearized Galerkin Finite Element Method for Nonlinear Wave Equations

High order hybrid asymptotic augmented finite volume methods for nonlinear degenerate wave equations

High order hybrid asymptotic augmented finite volume methods for nonlinear degenerate wave equations

A time‐domain boundary element method for the 3D dissipative wave equation: Case of Neumann problems

AbstractThe present article proposes a time‐domain boundary element method (TDBEM) for the three‐dimensional (3D) dissipative wave equation (DWE). Although the fundamental ingredients such as the Green's function for the 3D DWE have been known for a long time, the details of formulation and implementation for such a 3D TDBEM have been unreported yet to the author's best knowledge. The present formulation is performed truly in time domain on the basis of the time‐dependent Green's function and results in a marching‐on‐in‐time fashion. The main concern is in the evaluation of the boundary integrals. For this regard, weakly‐ and removable‐singularities are carefully treated. The proposed TDBEM is checked through the numerical examples whose solutions can be obtained semi‐analytically by means of the inverse Laplace transform. The results of the present TDBEM are satisfactory to validate its formulation and implementation for Neumann problems. On the other hand, the present formulation based on the ordinary boundary integral equation (BIE) is unstable for Dirichlet problems. The numerical analyses for the non‐dissipative case imply that the instability issue can be partially resolved by using the Burton–Miller BIE even in the dissipative case.

Time-explicit hybrid high-order method for the nonlinear acoustic wave equation

We devise a fully explicit scheme for the nonlinear acoustic wave equation in its second-order formulation in time, using the HHO method for space discretization and the leapfrog scheme for time integration. The key idea for the explicitation is an iterative procedure to approximate at each time step the static nonlinear coupling between the cell and face unknowns. This procedure is based on a splitting of the HHO stabilization operator and, in the linear case, is proved to converge for a large enough weight of the stabilization uniformly in the mesh size. Increasing the stabilization weight turns out to have a moderate impact on the CFL condition. The numerical experiments demonstrate the computational efficiency of the splitting procedure compared to a semi-implicit scheme for the static coupling between cell and face unknowns.

Identification and extraction of lateral target signals in tunnel geological prediction with the Karhunen-Loéve beamforming method

Before the construction of a tunnel, effective and accurate advanced prospecting techniques are required to detect unexpected geological heterogeneity around the tunnel. However, because the data collected during tunnel advance prediction include waves reflected from different directions and a large amount of environmental noise, the seismic records collected are extremely complex, and it is difficult to extract the target signals from the lateral area of the tunnel effectively. The beamforming method can create a focused wave front through the stack of the wavefield, improving the data quality of the target signal. This paper incorporates the Karhunen-Loéve method in the beamforming procedure to extract the lateral signal of the tunnel. First, the test principle of Karhunen-Loéve beamforming (KLBF) based on the tunnel seismic prediction (TSP) observation system is discussed, and the corresponding delay parameters are estimated according to different application conditions. In addition, ray tracing and three-dimensional wave equation methods are used to carry out numerical simulation research on the models of faults, karst caves and underground rivers. Finally, the corresponding KLBF method can be used to improve the quality of effective signals by different delay time calculation methods, which can better identify and extract the lateral reflected signals.

A space-time domain RBF method for 2D wave equations

In the present study, we demonstrate the feasibility to reveal the numerical solution of the multi-dimensional wave equations. A simple semi-analytical meshless method was proposed to obtain the numerical solution of the wave equation with a newly-proposed space-time radial basis function to enhance the numerical stability. The wave equation was discretized into equivalent algebraic equations. By specifying boundary and initial conditions, the wave propagation in a two-dimensional domain can be virtually reconstructed. Our results exhibit that the semi-analytical meshless method is suitable and efficient for solving multi-dimensional wave equations.

Energy-preserving splitting finite element method for nonlinear stochastic space-fractional wave equations with multiplicative noise

It is well-known that numerical methods that preserve physical quantities of original system often yield physically reasonable results and possess better numerical stability in long-time simulations. In this paper, we propose a methodology of constructing efficient fully-discrete method preserving averaged energy evolution law for nonlinear stochastic space-fractional wave equations with multiplicative noise. Specifically, the space variable is firstly discretized by Galerkin finite element method, and then the splitting technique and averaged vector field method are employed for the time integration of the resulting spatial semi-discrete scheme. We prove that the proposed fully-discrete method preserves the averaged energy evolution law in the discrete level. Numerical experiments are given to verify the theoretical result.

Weighted energy method for semilinear wave equations with time-dependent damping

Of concern is the energy decay property of solutions to wave equations with time-dependent damping. A reasonable class of damping coefficients for the framework of weighted energy methods is proposed, which contains not only the model of “effective” damping (1+t)^{-beta }(-1le beta <1), but also non-differentiable functions with a suitable behavior at trightarrow infty . As an application of the weighted energy estimate, global existence for the corresponding semilinear wave equation is discussed.

Q-Compensated Gaussian Beam Migration under the Condition of Irregular Surface

The viscosity of actual underground media can cause amplitude attenuation and phase distortion of seismic waves. When seismic images are processed assuming elastic media, the imaging accuracy for the deep reflective layer is often reduced. If this attenuation effect is compensated, the imaging quality of the seismic data can be significantly improved. Q-compensated Gaussian beam migration (Q-GBM) is an effective seismic imaging method for viscous media, and it has the advantages of both wave equation and ray-based Q-compensated imaging methods. This study develops a Q-GBM method in visco-acoustic media with an irregular surface. Initially, the basic principles of Gaussian beam in visco-acoustic media are introduced. Then, by correcting the complex-value time of the Gaussian beam in visco-acoustic media, energy compensation and phase correction are carried out for the forward continuation wavefield at the seismic source of the irregular surface and the reverse continuation wavefield at the beam center, which effectively compensates the absorption and attenuation effects of visco-acoustic media on the seismic wavefield. Further, a Q-GBM method under the irregular surface is proposed using cross-correlation imaging conditions. Through migration tests for three numerical models of visco-acoustic media with irregular surfaces, it is verified that our method is an effective depth domain imaging technique for seismic data in visco-acoustic media under the condition of irregular surfaces.

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

Fast Computing Approaches Based on a Bilinear Pseudo-Spectral Method for Nonlinear Acoustic Wave Equations

Fast Computing Approaches Based on a Bilinear Pseudo-Spectral Method for Nonlinear Acoustic Wave Equations

A fixed-point iteration method for high frequency vector wave equations

For numerically solving the high frequency vector wave equations, we present a simple approach based on fixed point iterations, where the problem is transferred into a fixed-point problem related to an exponential operator. The associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for prescribed termination criteria. For the sub-operator that is related to non-constant relative permeability, the Krylov subspace method or Taylor expansion is adapted for approximating its exponential. Furthermore, the Anderson acceleration is incorporated to accelerate the convergence of the fixed-point iterations. Numerical experiments are presented to demonstrate the accuracy and efficiency of the method.

Embedding antennas with tilted beam patterns into parabolic wave equation‐based models for tunnel propagation

AbstractAccurate radio wave prediction models are necessary for the innovation of mobile communication technology and the deployment of wireless sensor networks in railways. The parabolic wave equation (PWE) method is one commonly used method to predict the electromagnetic characteristic of long railway tunnels. However, the advent of novel antennas and technologies (e.g., beam steering, beamforming) has posed new challenges to PWE‐based models such as effective means of taking tilted beam patterns into account. This study addresses such a challenge by using tilted Gaussian beams to embed antennas with oriented patterns or tilted beam directions into PWE‐based models. The proposed approach does not reduce the simulation efficiency of PWE itself. The validity and practicability of the technique are demonstrated by comparing against ray‐tracing as well as experimental measurements in various tunnel environments.

Energy-conserving successive multi-stage method for the linear wave equation with forcing terms

We propose a high-order time-discretized method for a non-homogeneous linear wave equation with a forcing term. The method conserves the accumulated discrete energy with the external term. We provide detailed proofs of unique solvability and unconditional energy conservation of the proposed successive multi-stage (SMS) method. We also present reduced order conditions up to the fourth order with aid of some important algebraic identities from the features of the SMS methods. We demonstrate the accuracy and stability of the SMS methods using numerical experiments. In addition, to show the applicability of the proposed method, we extend the method to solve quasi-linear wave equations and provide numerical simulations for sine-Gordon and Boussinesq-type equations.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

L2 estimates for weak Galerkin finite element methods for second-order wave equations with polygonal meshes

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L2 norm is shown to hold as O(hk+1+τ2), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

Research Status and Progress of Multiples Suppression Based on Radon Transform

The existence of multiple waves in seismic data has a substantial impact on earthquake imaging, inversion, and interpretation results. As a result, multiples are often suppressed as noise prior to seismic data preattack processing. Currently, there are two primary methods for suppressing multiple reflections in seismic data. One is a filtering technique based on geometric differences in the seismic waves, while the other is based on wave equation methods. The Radon transform, which suppresses multiple reflections, belongs to the class of geometric seismic diffraction filtering techniques. In the process of seismic data processing, it effectively separates multiple reflections from seismic data by utilizing the characteristic differences between primary and multiple reflections, thereby achieving noise attenuation and improving the signal-to-noise ratio of the data. This article primarily investigates the current state of methods for suppressing multiple reflections based on the Radon transform, both domestically and internationally. By introducing various forms and fundamental principles of the Radon transform, this study analyzes and compares the adaptability and pros and cons of each type of Radon transform. The high-precision Radon transform can effectively address the sawtooth phenomenon and energy dispersion issues that arise in the least-squares Radon transform. It also highlights the challenges associated with suppressing multiple reflections in the Radon transform and provides a glimpse into future developments.

Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.

Application of modified exp-function method for strain wave equation for finding analytical solutions

The strain wave equation in micro-structured solids, which is significant in solid physics, is offered for consideration in this work. This is accomplished using the modified exp-function method, which is one of the most powerful ways for constructing abundantly distinct, accurate solutions to nonlinear partial differential equations. Wave propagation in microstructured solids is based on the structure of the strain wave equation. As a result, we were able to obtain a variety of new analytical solitary wave solutions like hyperbolic and complex function solutions with free parameters. Additionally, 2D and 3D plots are graphed for each of the solutions found in order to better comprehend their physical interpretation.

Numerical Analysis of Target Scattering Characteristics Based on Singularity Expansion Method

Based on the physical meaning of wave equation and Singularity Expansion Method, a numerical solution method is designed to extract the pole distributions of infinite cylinder, sphere and ellipsoid. By analysing the wave equation and boundary conditions of the target scattering characteristics, the extractions of poles of target scattering field are transformed into the solution of Hankel function and its derivative roots. Then, the T-matrix method is improved to make the numerical solution process of roots more accurate and faster. The numerical results are compared with strict analytical solutions and experimental results to fully verify the correctness of the method mentioned in this paper this study provides a basis for understanding the scattering mechanism of surface waves, and also provides a method and idea for solving problems such as acoustic scattering and electromagnetic scattering in engineering practice.