Two-dimensional Wave Equation Research Articles

Global Dynamics for the Two-dimensional Stochastic Nonlinear Wave Equations

AbstractWe study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (1) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (2) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain’s invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.

Infinitely many non-uniqueness examples for Cauchy problems of the two-dimensional wave and Schrödinger equations

In 1963, Kumano-go presented one non-uniqueness example for the two-dimensional wave equation with a time-dependent potential. We construct infinitely many non-uniqueness examples with different wave numbers at infinity for Cauchy problems of the two-dimensional wave equation and Schrödinger equation as a generalization of the construction by Kumano-go.

Derivation of Second Order Partial Differential Equation Indicating Wave and Heat Equation through the Use of the Navier Stoke’s Equation for Unsteady and Incompressible Flow

In this paper, we have tried to approach the concepts of two-dimensional wave equation and one dimensional heat equation through the means of the Navier Stoke’s equation for unsteady and incompressible flow. Our pursuit to do so has been supported with ample justifications and analytic discussions. The strong relation shared by the fluid dynamics, wave mechanics and heat flow has been brought to light through our attempts.

Vibration and horizontal directivity analysis of transmitting transducer for acoustic logging while drilling

Abstract Radially polarized open-cylindrical piezoelectric transducers are widely used in the field of acoustic logging while drilling (LWD). Unlike the wireline logging transducer, the structure of the acoustic LWD transducer is an open structure; in this case, its radial vibration is accompanied by apparent circumferential vibration. In this paper, based on the two-dimensional wave equations and electrostatic charge equation, according to the free boundary conditions of the transducer, the resonance frequency equation of radial-circumferential coupled vibration for the acoustic LWD transducer is obtained. The vibration modal and resonance frequency for transducers of several typical sizes are simulated using COMSOL software. The results show that when the geometrical size of the transducer satisfies certain conditions, the theoretical calculation and the numerical analysis are in good agreement and the relative error is controlled below 3%. Meanwhile, the horizontal directivity of the LWD transducer after actual installation is discussed, and it is found that adding appropriate coupling materials can improve the monopole sound field radiation. So, it is expected that this work can serve as a reference for the acoustic LWD transducer design and install.

The Method of Lines Analysis of TE Mode Propagation in Silica based Optical Directional Couplers

Optical directional couplers fabricated using planar light wave circuit (PLC) technology are versatile tools in integrated photonics devices. They have the advantages of small size, high consistency, ability for high volume production, and excellent possibility to be integrated with electronics circuits. Optical waveguide couplers are mainly utilized as power dividers, optical switches, and wavelength division multiplexers/de-multiplexers (WDM). A number of methods have been used to analyze directional couplers, such as coupled mode theory (CMT), the beam propagation method (BPM), the method of lines (MoL), finite-difference methods (FDM), and finite element methods (FEM). Among these numerical approaches, MoL is the simplest method to analyze mode propagation inside directional couplers because it has the advantages of very fast convergence and accurate solutions for one-dimensional structures. The objective of this study was to analyze the propagation of TE modes in optical directional couplers by using MoL. The parameters used, i.e. waveguide width, refractive index, and wavelength, were taken from the characteristics of silica-on-silicon directional couplers that were used in fabrication. MoL is considered a special finite-difference method, which discretizes a one- or two-dimensional wave equation in the transverse direction and uses an analytical solution for the propagation directions. Basically, MoL is a semi analytical numerical method with the advantages of numerical stability, computational efficiency, and calculation time reduction. Further, we explored the possibility of using directional couplers as optical switching devices.

Asymptotics of the Head Wave in the Cauchy Problem for a Difference Scheme Corresponding to the Two-Dimensional Wave Equation with Localized Initial Data

Asymptotics of the Head Wave in the Cauchy Problem for a Difference Scheme Corresponding to the Two-Dimensional Wave Equation with Localized Initial Data

The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

Spatially Quasi-Periodic Water Waves of Infinite Depth

We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

Global well posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain

We study the two-dimensional wave equation with cubic nonlinearity posed on R2 with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for initial data in Hs, s>45.

A compact scheme for two-dimensional nonlinear time fractional wave equations

Based on the equivalent integro-differential form of the considered problem, a numerical approach to solving the two-dimensional nonlinear time fractional wave equations (NTFWEs) is considered in this paper. To this end, an alternating direction implicit (ADI) numerical scheme is derived. The scheme is established by combining the second-order convolution quadrature formula and Crank–Nicolson technique in time and a fourth-order difference approach in space. The convergence and unconditional stability of the proposed compact ADI scheme are strictly discussed after a concise solvability analysis. A numerical example is shown to demonstrate the theoretical analysis.

Exact Bounded Boundary Controllability to Rest for the Two-Dimensional Wave Equation

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in finite time.

Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha > 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha $. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$. These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

A GPU implementation of least-squares reverse time migration

Least-squares reverse time migration (LSRTM) is a seismic imaging method that can provide higher-resolution image of the subsurface structures compared to other methods. However, LSRTM is computationally expensive. To reduce the computational time of LSRTM, GPU can be utilized. This leads to the objective of this work which is to develop a GPU implementation of LSRTM. In this work, the two-dimensional first-order acoustic wave equations are solved using the second-order finite difference on a staggered grid and a perfectly matched layer is used as an absorbing boundary condition. The adjoint-state method is used to compute the gradient of the objective function. A linear conjugate gradient method is used to minimize the objective function. Both forward- and backward-propagation of wavefields using the finite-difference method are performed on a single GPU using the NVIDIA CUDA library. For a verification purpose, the GPU program of LSRTM was applied to a synthetic data set generated from the Marmousi model. Numerical results show that LSRTM can provide an image with a higher resolution of subsurface structure compared to a conventional RTM image. For a computational cost issue, the GPU-version of LSRTM is significantly faster than the serial CPU-version of LSRTM.

Asymptotics of the Head Wave in the Cauchy Problem for a Difference Scheme Corresponding to the Two-Dimensional Wave Equation with Localized Initial Data

В работе строится асимптотика головной волны задачи Коши для псевдодифференциального уравнения, отвечающего разностной схеме для волнового уравнения с локализованными начальными данными. В зависимости от соотношения параметров задачи у головной волны возникают дисперсионные эффекты. В случае сильных дисперсионнных эффектов строится равномерная асимптотика головной волны в достаточно большой окрестности переднего фронта. Более того, такую асимптотику можно представить с помощью тэта-функций Якоби в случае, когда начальная функция имеет вид гауссовой экспоненты. Библиография: 14 названий.

A Deep Learning Approach for Predicting Spatiotemporal Dynamics From Sparsely Observed Data

In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs). We propose a deep learning framework that learns the underlying dynamics and predicts its evolution using sparsely distributed data sites. Deep learning has shown promising results in modeling physical dynamics in recent years. However, most of the existing deep learning methods for modeling physical dynamics either focus on solving known PDEs or require data in a dense grid when the governing PDEs are unknown. In contrast, our method focuses on learning prediction models for unknown PDE-driven dynamics only from sparsely observed data. The proposed method is spatial dimension-independent and geometrically flexible. We demonstrate our method in the forecasting task for the two-dimensional wave equation and the Burgers-Fisher equation in multiple geometries with different boundary conditions, and the ten-dimensional heat equation.

Classical solutions for a class of IVP for nonlinear two-dimensional wave equations via new fixed point approach

This paper devoted to research a classical solutions for a range of two-dimensional problems associated with the propagation of nonlinear waves subjected with a class of initial value problems of nonlinear equations in hyperbolic type. A new fixed point approach is used to prove the existence of classical positive solutions of the considered IVP. The results in the manuscript are provided with examples.

Moving Three Collinear Griffith Cracks at Orthotropic Interface

This work deals with the interaction of P-waves between a moving central crack and a pair of outer cracks situated at the interface of an orthotropic layer and an elastic half-space. Initially, we considered a two-dimensional elastic wave equation in orthotropic medium. The Fourier transform has been applied to convert the basic problem to solve the set of four integral equations. These set of integral equations have been solved to to get the analytical expressions for the stress intensity factor (SIF) and crack opening displacements (COD) by using the finite Hilbert transform technique and Cooke’s result. The main objective of this work is to investigate the dynamic stress intensity factors and crack opening displacement at the tips of the cracks. The aims of the study of these physical quantities (SIF, COD) is the prediction of possible arrest of the cracks within a certain range of crack velocity by monitoring applied load. SIF and COD have been depicted graphically for various types of orthotropic materials. We presented a parametric study to explore the influence of crack growing and propagation. This result is very much applicable in bridges, roads, and buildings fractures.

An explicit fourth-order compact difference scheme for solving the 2D wave equation

In this paper, an explicit fourth-order compact (EFOC) difference scheme is proposed for solving the two-dimensional(2D) wave equation. The truncation error of the EFOC scheme is O({tau ^{4}} + {tau ^{2}}{h^{2}} + {h^{4}}), i.e., the scheme has an overall fourth-order accuracy in both time and space. Because the scheme is explicit, it does not need any iterative processes. Afterwards, the stability condition of the scheme is obtained by using the Fourier analysis method, which has a wider stability range than other explicit or alternation direction implicit (ADI) schemes. Finally, some numerical experiments are carried out to verify the accuracy and stability of the present scheme.

High-order point-value enhanced finite volume method for two-dimensional hyperbolic equations on unstructured meshes

Presently, an approximate delta function (ADF) is defined in multi-dimensional space in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand spanning a finite domain. By using this ADF over a two-dimensional unstructured mesh, a compact high-order point-value enhanced finite volume method (PFV) is constructed. The corresponding scheme is capable of storing and updating cell-averaged values inside each element as well as the unknown quantities and their derivatives (when needed) on vertex points. The sharing of vertex information with surrounding elements reduces the number of degrees of freedom relative to other compact and high-order methods of comparable convergence rate. To ensure conservation, cell-averaged values are updated using an identical approach to the one traditionally followed in the finite volume method. To verify the performance of the PFV scheme at successive orders, its accuracy and stability are vetted using benchmark problems associated with the two-dimensional wave and Euler equations encompassing a total of six standard test cases.

A parallel hybrid implementation of the 2D acoustic wave equation

Abstract In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.