Boundary element calculation of shallow water waves

A boundary element method is proposed for studying periodic shallow water problems. The numerical model is based on the shallow water equation. The key feature of this method is that the boundary integral equations are derived using the weighted residual method and the fundamental solutions for shallow water wave problems are obtained by solving the simultaneous singular equations. The accuracy of this method is studied for the wave reflection problem in a rectangular tank. As a result of this test, it has been shown that the number of element divisions and the distribution of nodes are significant to the accuracy. For numerical examples of external problems, the wave diffraction problems due to single cylindrical, double cylindrical and plate obstructions are analysed and compared with the exact and other numerical solutions. Relatively accurate solutions are obtained.     

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Her-mite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection- diffusion problem) and KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Improved non-singular local boundary integral equation method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM),singularities in the local boundary integrals need to be treated specially. In the current paper,local integral equations are adopted for the nodes inside the domain trod moving least square approximation (MLSA) for the nodes on the global boundary,thus singularities will not occur in the new al- gorithm.At the same time,approximation errors of boundary integrals are reduced significantly.As applications and numerical tests,Laplace equation and Helmholtz equa- tion problems are considered and excellent numerical results are obtained.Furthermore, when solving the Hehnholtz problems,the modified basis functions with wave solutions are adapted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.     

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations

In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy.     

FEM-FDM coupled liquefaction analysis of a porous soil using an elasto-plastic model

The phenomenon of liquefaction is one of the most important subjects in Earthquake Engineering and Coastal Engineering. In the present study, the governing equations of such coupling problems as soil skeleton and pore water are obtained through application of the two-phase mixture theory. Using au-p (displacement of the solid phase-pore water pressure) formulation, a simple and practical numerical method for the liquefaction analysis is formulated. The finite difference method (FDM) is used for the spatial discretization of the continuity equation to define the pore water pressure at the center of the element, while the finite element method (FEM) is used for the spatial discretization of the equilibrium equation. FEM-FDM coupled analysis succeeds in reducing the degrees of freedom in the descretized equations. The accuracy of the proposed numerical method is addressed through a comparison of the numerical results and the analytical solutions for the transient response of saturated porous solids. An elasto-plastic constitutive model based on the non-linear kinematic hardening rule is formulated to describe the stress-strain behavior of granular materials under cyclic loading. Finally, the applicability of the proposed numerical method is examined. The following two numerical examples are analyzed in this study: (1) the behavior of seabed deposits under wave action, and (2) a numerical simulation of shaking table test of coal fly ash deposit.     

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

We describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth‐averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well‐balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run‐up of a solitary wave on a sloped beach. The finite‐volume finite‐element approach to solving the Serre equation was found to be accurate and robust.     

固体非傅立叶温度场的时域间断Galerkin有限元法

运用时域间断Galerkin有限元法[1],对高频非傅立叶热波动问题[2-3]进行分析。其主要特点是:取温度及温度的时间导数为基本未知量,对其分别进行3次Hermite插值和线性插值。在保证节点温度自动保持连续的基础上,温度的时间导数在离散时域存在间断。数值结果表明所提出的方法能够滤掉虚假的数值震荡,能够良好地模拟固体中的非傅立叶热波动行为。     

水下声波传播过程的时域间断Galerkin有限元方法

本文构建了声压波动方程的改进时域间断Galerkin有限元方法.传统时域连续有限元方法在计算高梯度、强间断特征水中声波传播问题时往往会出现虚假数值振荡现象,这些数值振荡会影响正常波动的计算精度.为了解决这一问题,本文通过引入人工阻尼的方式构建了改进的时域间断Galerkin有限元方法,并针对具有高梯度、强间断特征的多障碍物复杂边界和层合液体介质声传播问题进行了计算.计算结果表明,与传统时域连续方法如N ew mark方法计算结果对比,所发展方法能较好地消除高梯度和强间断声压力波传播过程中虚假的数值振荡,具有较高的计算精度.问题的求解为进一步流固声耦合问题的研究奠定了基础.     

Solving a fully nonlinear highly dispersive Boussinesq model with mesh‐less least square‐based finite difference method

Combining mesh‐less finite difference method and least square approximation, a new numerical model is developed for water wave propagation model in two horizontal dimensions. In the numerical formulation of the method, the approximation of the unknown functions and their derivatives are constructed on a set of nodes in a local circular‐shaped region. The Boussinesq equations studied in this paper is a fully nonlinear and highly dispersive model, which is composed of the exact boundary conditions and the truncated series expansion solution of the Laplace equation. The resultant system involves a sparse, unsymmetrical matrix to be solved at each time step of the simulation. Matrix solutions are studied to reduce the computing resource requirements and improve the efficiency and accuracy. The convergence properties of the present numerical method are investigated. Preliminary verifications are given for nonlinear wave shoaling problems; the numerical results agree well with experimental data available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite volume element method for analysis of unsteady reaction–diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

KdV方程孤立子及其相互作用的Petrov-Galerkin有限元数值模拟

本文给出一种Petrov-Galerkin有限元方法,并用以求Kdv方程各种初值问题的数值解,包括孤立波进波解,多个孤立于的相互作用,孤立子与振荡尾波等,所得结果与分析解及其它数值结果作了比较,表明本方法精度高、稳定性好,几乎没有高频伪振荡,计算程序简洁、明瞭,经济实用。     

Numerical computation of solitary wave solutions of the Rosenau equation

We construct numerically solitary wave solutions of the Rosenau equation using the Petviashvili iteration method. We first summarize the theoretical results available in the literature for the existence of solitary wave solutions. We then apply two numerical algorithms based on the Petviashvili method for solving the Rosenau equation with single or double power law nonlinearity. Numerical calculations rely on a uniform discretization of a finite computational domain. Through some numerical experiments we observe that the algorithm converges rapidly and it is robust to very general forms of the initial guess.     

A level set-based Eulerian approach for anisotropic wave propagation

The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of these wave propagations. In particular, our method is able to accurately reproduce the quasi-transverse, or quasi-S, waves with cusps, which form a class of multi-valued solutions. The level set formulation we use is borrowed from one for moving curves in three spatial dimensions, with the velocity fields for evolution following from the method of characteristics on the anisotropic eikonal equation. We present here our derivation of the algorithm and numerical results to illustrate its accuracy in different cases of anisotropic wave propagations related to seismic imaging.     

弹性薄板弯曲问题的边界轮廓法

导出了弹性薄板弯曲问题边界积分方程的另一种形式,基于这种方程,提出了平板弯曲问题的边界轮廓法,讨论了三次边界单元边界轮廓法的计算列式,并给出了计算内力的边界轮廓法方程。该法无需进行数值积分计算,完全避免了角点问题和奇异积分计算。给出的算例,与解析解相比较,证实该方法的有效性。     

波动数值模拟的一种显式方法——一维波动

本文提出了波动时域数值模拟的一种新的显式方法，并通过一维非规则网格节点递推公式的建立说明此方法。为了阐明应用此法构建高精度、稳定递推公式的可行性，详细论述了一维均匀网格标量波动数值模拟的精度和稳定性；提出了构建时空精度皆为 阶( 为正整数)的稳定递推公式的技术途径，并以构建二阶(M=1)和四阶(M=2)公式为例予以说明。最后，通过算例详细说明了本文理论结果。     

求解浅水波方程的并行物理信息神经网络算法

双曲守恒律方程是一类比较特殊的偏微分方程,其数值求解方法的研究一直是一个热点问题,一个显著特性是即使初始条件是光滑的,其解也可能会发展成间断。浅水波方程作为非线性双曲守恒律方程,由于间断解的存在,其精确求解存在很大困难。针对浅水波方程数值求解问题,本文基于PINN(Physics informed neural networks)反问题网络结构构造新的网络,构造的网络结构包括两个并行的神经网络,其中一个网络与已知状态数据(熵稳定格式加密求出)相关,另一个网络与方程本身相关。利用已知速度数据结合浅水波方程本身求解未知水深,最终通过一些数值算例验证网络的可行性。结果表明,新的网络结构可用于浅水波方程求解,利用速度数据可以较为精确地推算出水深。     

Higher order dispersive effects in regularized Boussinesq equation

In this paper, we consider the higher order Boussinesq (HBq) equation which models the bi-directional propagation of longitudinal waves in various continuous media. The equation contains the higher order effects of frequency dispersion. The present study is devoted to the numerical investigation of the HBq equation. For this aim a numerical scheme combining the Fourier pseudo-spectral method in space and a Runge–Kutta method in time is constructed. The convergence of semi-discrete scheme is proved in an appropriate Sobolev space. To investigate the higher order dispersive effects and nonlinear effects on the solutions of HBq equation, propagation of single solitary wave, head-on collision of solitary waves and blow-up solutions are considered.