A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations

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.

Error analysis of the spectral element method with Gauss–Lobatto–Legendre points for the acoustic wave equation in heterogeneous media

Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

A widely convergent generalized pulse-spectrum technique for the inversion of two-dimensional acoustic wave equation

The Galerkin solution of two-dimensional TE wave equation is derived from the two-dimensional vector wave equation by using basis (interpolation) functions defined in each element. And then, considering with the first-order absorbing boundary condition the time-domain recursive formulation is obtained by using Newmark-β method . Finally, a numerical example is given to illustrate the feasibility of the presented scheme.

Purpose Semilinear wave equations with different source terms describe acoustic wave motion in fluids, shock wave formation that decelerates fluid from supersonic to subsonic speeds and quenching phenomena in micro-electro mechanical systems devices with fluid mechanical applications. This paper aims to investigate the quenching behavior of numerical solutions for a two-dimensional semilinear wave equation with an inverse power law term. Design/methodology/approach The localized radial basis function-generated finite difference (RBF-FD) method is used for approximating numerical solutions in space, and the finite difference scheme is used for temporal discretization. A discrete energy analysis is conducted to evaluate the local stability of the developed numerical scheme. Findings The energy functional of the classical solution is defined. The numerical results demonstrate finite-time quenching, and the influence of various parameters is assessed through detailed numerical simulation. Originality/value An RBF-FD approach is applied to confront the quenching phenomena in one- and two-dimensional cases. Stability and the computational performance of the proposed numerical scheme are verified numerically. The impact of various parameters and domains on quenching time is studied in detail.

In investigating the unsteady transient pressurized pipe flows, including the water hammer (WH) phenomenon, accurate prediction of the pressure head history, mean velocity, and wall shear stress is crucial. In this regard, the quasi-two-dimensional (Q2D) approach is evergreen because it can consider the radial distribution of the local velocities and Reynolds stress and incorporate various types of turbulence models, whether frozen or time-varying. However, this study highlights the inherent deficiencies and limitations of the numerical solutions of the Q2D method in accurately predicting the attenuation and phase shape of WH oscillations in a simple pipeline. In addition, this paper presents a novel and comprehensive analytical solution employing the separation of variables method for the two-dimensional nonhomogenous damped wave equation, which is directly derived from the Q2D governing equations, providing significant advantages in precision and simplicity due to the lack of need for computational grid planning. The simulation results are validated against experimental data by calibrating the decay coefficients, i.e., damping and phase variation parameters, through genetic algorithm optimization.

Ultra-Weak Discontinuous Galerkin Methods for Two-Dimensional Diffusive-Viscous Wave 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.

The Cauchy problem is considered for the two-dimensional wave equation with velocity $c=\sqrt x_1$ on the half-plane $\{x_1\geq 0,x_2\}$, with initial data localized in a neighborhood of the point $(1,0)$. This problem serves as a model problem in the theory of beach run-up of long small-amplitude surface waves excited by a spatially localized instantaneous source. The asymptotic expansion of the solution is constructed with respect to a small parameter equal to the ratio of the source linear size to the distance from the $x_2$-axis (the shoreline). The construction involves Maslov’s canonical operator modified to cover the case of localized initial conditions. The relationship of the solution with the geometrical optics ray diagram corresponding to the problem is analyzed. The behavior of the solution near the $x_2$-axis is studied. Simple solution formulas are written out for special initial data.

We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations. Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.

<abstract><p>In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.</p></abstract>

The characteristics of unsteady flow motions have long been a serious concern in the study of various fluid dynamic and combustion problems. With the advancement of computer resources, numerical approaches to these problems appear to be feasible. The objective of this paper is to assess the accuracy of several numerical schemes for unsteady flow calculations. In the present study, Fourier error analysis is performed for various numerical schemes based on a two-dimensional wave equation. Four methods sieved from the error analysis are then adopted for further assessment. Model problems include unsteady quasi-one-dimensional inviscid flows, two-dimensional wave propagations, and unsteady two-dimensional inviscid flows. According to the comparison between numerical and exact solutions, although second-order upwind scheme captures the unsteady flow and wave motions quite well, it is relatively more dissipative than sixth-order central difference scheme. Among various numerical approaches tested in this paper, the best performed one is Runge-Kutta method for time integration and six-order central difference for spatial discretization.

Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients

A first, we used our knowledge of Fourier series to solve several interesting boundary value problems by the method of separation of variables. The success of our method depended to a large extent on the fact that the domains under consideration were easily described in Cartesian coordinates. In this paper/research we address problems where the domains are easly described in polar and cylindrical coordinates. Spesifically we consider boundary value problems for the wave, heat, Laplace and Poisson equation over disks or cylinders. Upon restating these problems in suitable coordinat systems and separating variables, we will encounter new ordinary differential equations, Bessel’s equation, whose solutions are called Bessel function in ways analogous to Fourier series expansions. The vibrations of the membrane are governed by the two-dimensional wave equation, which will be expressed in polar coordinantes, because these are the coordinates best suited to this problem. Finally, we will solve the two dimensional wave equation in polar coordinates (general case).