Two-dimensional Wave Equation Research Articles

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

Authentic fault models and dispersive tsunami simulations for outer-rise normal earthquakes in the southern Kuril Trench

The southern Kuril Trench is one of the most seismically active regions in the world. In this study, marine surveys and observations were performed to construct fault models for possible outer-rise earthquakes. Seismic and seafloor bathymetric surveys indicated that the dip angle of the outer-rise fault was approximately 50°–80°, with a strike that was slightly oblique to the axis of the Kuril Trench. The maximum fault length was estimated to be ~ 260 km. Based on these findings, we proposed 17 fault models, with moment magnitudes ranging from 7.2 to 8.4. To numerically simulate tsunami, we solved two-dimensional dispersive wave and three-dimensional Euler equations using the outer-rise fault models. The results of both simulations yielded identical predictions for tsunami with short-wavelength components, resulting in significant dispersive deformations in the open ocean. We also found that tsunami generated by outer-rise earthquakes were affected by refraction and diffraction because of the source location beyond the trench axis. These findings can improve future predictions of tsunami hazards.Graphical

Enhancing flood wave modelling of reservoir failure: a comparative study of structure-from-motion based 2D and 3D methodologies

Predicting flood wave propagation from reservoir failures is critical to practical flood hazard assessment and risk management. Flood waves are sensitive to topography, channel geometry, structures, and natural features along floodplain paths. Thus, the accuracy of flood wave modelling depends on how precisely those features are represented. This study introduces an enhancing approach to flood wave modelling by accurately representing three-dimensional objects in floodplains using the structure-from-motion (SfM). This method uses an unmanned aerial vehicle to capture topographic complexities and account for ground objects that impact flood propagation. Using the three-dimensional volume of fluid numerical approach significantly improves an enhanced representation of turbulent flow dynamics and computational efficiency, especially in handling large topography datasets. Reproductions from this enhanced three-dimensional approach were validated against recent reservoir failure observations and contrasted with traditional two-dimensional models. The results revealed that the suggested three-dimensional methodology achieved a significant 84.4% reproducibility when juxtaposed with actual inundation traces. It was 35.5%p more accurate than the two-dimensional diffusion wave equation (DWE) and 17.1%p more than the shallow water equation (SWE) methods in predicting flood waves. This suggests that the reproducibility of the DWE and SWE decreases compared to the three-dimensional approach when considering more complex floodplains. These results demonstrate that three-dimensional flood wave analysis with the SfM methodology is optimal for effectively minimising topographic and flood wave reproduction errors across extensive areas. This dual reduction in errors significantly enhances the reliability of flood hazard assessments and improves risk management by providing more precise and realistic predictions of flood waves.

Embedded exponential Runge–Kutta–Nyström methods for highly oscillatory Hamiltonian systems

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O(N2) to O(Nlog⁡N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

On the utilization of the adjoint method in microwave tomography.

In microwave imaging, the adjoint method is widely used for the efficient calculation of the update direction, which is then used to update the unknown model parameter. However, the utilization and the formulation of the adjoint method differ significantly depending on the imaging scenario and the applied optimization algorithm. Because of the problem-specific nature of the adjoint formulations, the dissimilarities between the adjoint calculations may be overlooked. Here, we have classified the adjoint method formulations into two groups: the direct and indirect methods. The direct method involves calculating the derivative of the cost function, whereas, in the indirect method, the derivative of the predicted data is calculated. In this review, the direct and indirect adjoint methods are presented, compared, and discussed. The formulations are explicitly derived using the two-dimensional wave equation in frequency and time domains. Finite-difference time-domain simulations are conducted to show the different uses of the adjoint methods for both single source-multiple receiver, and multiple transceiver scenarios. This study demonstrated that an appropriate adjoint method selection is significant to achieve improved computational efficiency for the applied optimization algorithm.

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

A new ADI-IIM scheme for solving two-dimensional wave equation with discontinuous coefficients

<p>A new alternating direction implicit immersed interface method (ADI-IIM) scheme was developed to solve the two-dimensional wave equation with discontinuous coefficients and sources. The alternating direction implicit (ADI) method was equipped with the immersed interface method (IIM) to recover the accuracy as well as maintaining the stability. Numerical experiments were carried out to verify the unconditional stability and the second-order accuracy both in time and space of the proposed scheme.</p>

A Hard-Constraint Wide-Body Physics-Informed Neural Network Model for Solving Multiple Cases in Forward Problems for Partial Differential Equations

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs.

Linear filtration waves in a layered heterogeneous formation

Solutions to the problem of a filtration-wave pressure field in a layered inhomogeneous medium with a pressure pulse specified on the left boundary are obtained. The problem statement contains a two-dimensional wave equation in the central layer, a wave equation that takes into account the predominance of vertical motion in the environment, symmetry conditions in the center of the formation, equality of pressures and flows at the interfaces, the absence of disturbances of the filtration wave field at the initial moment of time and at an infinite distance from the source of disturbance. The article found an exact solution to the problem obtained using the Laplace–Carson and the Fourier transform of sine, an asymptotic solution and an exact solution for the special case of a homogeneous medium. It is shown that when the formal parameter tends to zero, the exact solution coincides with the main asymptotic approximation. The tendency of the thickness of the central layer to infinity reduces the solution for a layered inhomogeneous medium to an expression describing the pressure fields in a homogeneous medium. Spatiotemporal dependencies describing the dynamics of the pressure pulse in layered inhomogeneous and homogeneous media were constructed, and the general properties and features of linear filtration waves in layered inhomogeneous and homogeneous layers were studied.

Physics-constrained neural networks for half-space seismic wave modeling

Forward modeling of seismic waves using physics-informed neural networks (PINNs) has attracted much attention. However, a notable challenge arises when modeling seismic wave propagation in large domains (i.e., a half-space), PINNs may encounter the issue of "soft constraint failure". To address this problem, we propose a novel framework called physics-constrained neural networks (PCNNs) specifically designed for modeling seismic wave propagation in a half-space. The method of images is incorporated to effectively implement the free stress boundary conditions of the Earth's surface, leading to the successful propagation of plane waves and cylindrical waves in a half-space. We analyze the training dynamics of neural networks when solving two-dimensional (2D) wave equations from the neural tangent kernel (NTK) perspective. An adaptive training algorithm is introduced to mitigate the unbalanced gradient flow dynamics of the different components of the loss function of PINNs/PCNNs. Furthermore, to tackle the complex behavior of seismic waves in layered media, a sequential training strategy is considered to enhance network scalability and solution accuracy. The results of numerical experiments demonstrate the accuracy and effectiveness of our approach.

Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev–Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

A Splitting Nearly-Analytic Symplectic Partitioned Runge–Kutta Method for Solving 2D Elastic Wave Equations

In this paper, we present a new splitting nearly-analytic symplectic partitioned Runge–Kutta (SNSPRK) method for the two-dimensional (2D) elastic wave equations. It is an extension to elastic wave equation of our recent work on the locally one-dimensional nearly-analytic symplectic partitioned Runge–Kutta (LOD-NSPRK) method for the 2D acoustic wave equations. The method is based on the spatial differential operator-split technique, in which the resulting spatial discrete matrices are symmetric unlike the conventional nearly-analytic symplectic partitioned Runge–Kutta (NSPRK) method. The stability condition is given, which has more relax restriction for the time step than the NSPRK schemes. To show the performance of the new method, numerical experiments are given. Numerical results illustrate that the SNSPRK method has better long-term calculation capability as time proceeds than the NSPRK method.

Investigate the dynamic nature of soliton solutions and bifurcation analysis to a new generalized two-dimensional nonlinear wave equation with its stability

In this current research, we investigate a new generalized two-dimensional nonlinear wave equation of engineering physics using two versatile approaches. Nonlinear wave models are very important in that they play a key role in modeling diverse occurrences in cosmology, solid mechanics and fluid dynamics, astrophysics, mathematical physics and engineering. The research work is based on three steps. Firstly, soliton solutions are secured via the modified Sardar sub-equation method (MSSEM) and the generalized exponential rational function method (GERFM). Secondly, for bifurcation analysis phase portraits of an equation are erected. Also, using the Galilean transformation, the given equation plane dynamical system is erected and all possible phase portraits are provided. Lastly, the modulation instability of the equation is examined. In order to better comprehend how the nonlinear wave equation depicts physical processes, the wave dynamics of the solutions are displayed to give the results a more physical perspective. The use of these findings in physics, engineering and complex media is essential.

Adding integrable defects to the Boussinesq equation

The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect contributes to energy and momentum to ensure their conservation, despite the presence of discontinuities and the explicit breaking of translation invariance, leads to sewing conditions relating the two fields and their derivatives on either side of the defect. The manner in which several types of soliton solutions to the Boussinesq equation are affected by the defect is explored and reveals new effects that have not been observed in other integrable systems, such as the possibility of a soliton reflecting from a defect or of a defect decaying into one or two solitons.

First Boundary Value Problem for the Two-Dimensional Wave Equation

First Boundary Value Problem for the Two-Dimensional Wave Equation

Uniform estimates for 2D quasilinear wave

We consider two-dimensional quasilinear wave equations with standard null-type quadratic nonlinearities. In 2001 Alinhac proved that such systems possess global in time solutions for compactly supported initial data with sufficiently small Sobolev norm. The highest norm of the constructed solution grows polynomially in time. In this work we develop a new strategy and prove uniform boundedness of the highest order norm of the solution for all time.

Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media

This article investigates a new generalized two-dimensional nonlinear wave equation of engineering physics with various applications in the fields of sciences and engineering. In this study, shock wave and solitary wave solutions were secured via the sine–Gordon technique. Moreover, various new group invariants along-side exact classical results of the equation were achieved through the utilization of Lie group theoretic techniques. Some of the solutions are gained with regards to Weierstrass functions, complex soliton, topological kink soliton as well as singular soliton. Besides, several algebraic and other solitary-wave-type solutions are obtained. Wave dynamics of the solutions are plotted to give more physical meanings to the obtained results and have a better knowledge of what the nonlinear wave equation represents in terms of physical phenomena. Application of the secured results in engineering (signal processing), physics (electromagnetism) and complex media are presented.

Physics-informed neural networks for transcranial ultrasound wave propagation

Transcranial ultrasound imaging has been playing an increasingly important role in the non-invasive treatment of brain disorders. However, the conventional mesh-based numerical wave solvers, which are an integral part of imaging algorithms, suffer from limitations such as high computational cost and discretization error in predicting the wavefield passing through the skull. In this paper, we explore the use of physics-informed neural networks (PINNs) for predicting the transcranial ultrasound wave propagation. The wave equation, two sets of time snapshots data and a boundary condition (BC) are embedded as physical constraints in the loss function during training. The proposed approach has been validated by solving the two-dimensional (2D) acoustic wave equation under three increasingly complex spatially varying velocity models. Our cases demonstrate that due to the meshless nature of PINNs, they can be flexibly applied to different wave equations and types of BCs. By adding physics constraints to the loss function, PINNs can predict wavefields far outside the training data, providing ideas for improving the generalization capability of existing deep learning methods. The proposed approach offers exciting perspectives because of the powerful framework and simple implementation. We conclude with a summary of the strengths, limitations and further research directions of this work.