Local discontinuous Galerkin approximations to fractional Bagley-Torvik equation

In this research, numerical approximation to fractional Bagley-Torvik equation as an important model arising in fluid mechanics is investigated. Our discretization algorithm is based on the local discontinuous Galerkin (LDG) schemes along with using the natural upwind fluxes, which enables us to solve the model problem element by element. This means that we require to solve a low-order system of equations in each subinterval, hence avoiding the need for a full global solution. The proposed schemes are tested on a range of initial- and boundary-value problems including a variable coefficient example, a nonsmooth problem, and some oscillatory test cases with exact solutions. Also, the validation of the proposed methods was compared with those obtained available existing computational procedures. Overall, it was found that LDG methods indicated highly satisfactory performance with comparatively lower degree of polynomials and number of elements compared with other numerical models.     

三维电阻抗成像的体积元方法的数值模拟和分析

电阻抗成像是一类椭圆方程反问题,本文在三维区域上对其进行数值模拟和分析.对于椭圆方程Neumann边值正问题,本文提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了误差泛函的Jacobi矩阵的计算公式,利用体积元格式的对称性和特殊的电流基向量,将每次迭代中需要求解的正问题的个数降到最低.一系列数值实验的结果验证了数学模型的可靠性和算法的可行性.本文所提出的这些方法,已成功应用于三维电阻抗成像的实际数值模拟.     

A quadratic b-spline based isogeometric analysis of transient wave propagation problems with implicit time integration method

A uniform quadratic b-spline isogeometric element is exclusively considered for wave propagation problem with the use of desirable implicit time integration scheme. A generalized numerical algorithm is proposed for dispersion analysis of one-dimensional (1-D) and two-dimensional (2-D) wave propagation problems where the quantified influence of the defined CFL number on wave velocity error is analyzed and obtained. Meanwhile, the optimal CFL (Courant–Friedrichs–Lewy) number for the proposed 1-D and 2-D problems is suggested. Four representative numerical simulations confirm the effectiveness of the proposed method and the correctness of dispersion analysis when appropriate spatial element size and time increment are adopted. The desirable computation efficiency of the proposed isogeometric method was confirmed by conducting time cost and calculation accuracy analysis of a 2-D numerical example where the referred FEM was also tested for comparison.     

An optimal‐order estimate for MMOC‐MFEM approximations to porous medium flow

Mathematical models used to describe porous medium flow lead to coupled systems of time‐dependent partial differential equations. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid‐orientation effect. The MMOC‐MFEM time‐stepping procedure, in which the modified method of characteristics (MMOC) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this article we prove an optimal‐order error estimate for a family of MMOC‐MFEM approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

动载荷识别的非迭代法研究

为了快速准确地识别结构在复杂环境下的承载状态，基于有限元法和Newmark-β法提出了一种非迭代反演方法，并用于识别结构上施加的动载荷.通过探寻测量信息与待演参量之间的关系，建立误差函数，根据最小二乘法实现动载荷的直接识别无需迭代，其中对待反演的分布载荷实施基函数展开，以提高算法的抗不适定性.同时奇异值分解法被用来求解病态方程组.数值算例分别讨论了测量噪声、测点数量、基函数展开、测点位置和不同时间步长对反演结果的影响，结果显示该方法在识别动载荷时具有较高的精度和效率.     

Optimal balance between mass and smoothed stiffness in simulation of acoustic problems

The Finite Element Method (FEM) is known to behave overly-stiff, which leads to an imbalance between the mass and stiffness matrices within discretized systems. In this work, for the first time, a model is developed that provides optimal balance between discretized mass and smoothed stiffness—the mass-redistributed alpha finite element method (MR-αFEM). This new method improves on the computational efficiency of the FEM and Smoothed Finite Element Methods (S-FEM). The rigorous research conducted ensures that stiffness with the parameter, α, optimally matches the mass with a flexible integration point, q. The optimal balance system significantly reduces the dispersion error of acoustic problems, including those of single and multi-fluids in both time and frequency domains. The excellent properties of the proposed MR-αFEM are validated using theoretical analyses and numerical examples.     

Two-Level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics

In this paper we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with compact fourth-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large linear systems of equations. We propose an iterative two-grid method where the coarse grid problem is solved inexactly. A single cycle of this method is used as a variable preconditioner for a flexible Krylov subspace method. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application. The proposed numerical method allows us to solve wave propagation problems with single or multiple sources even at high frequencies on a reasonable number of cores of a distributed memory cluster.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

Quadratic trigonometric b-spline galerkin methods for the regularized long wave equation

In this study, a numerical solution of the Regularized Long Wave (RLW) equation is obtained using Galerkin finite element method, based on two and three steps Adams Moulton method for the time integration and quadratic trigonometric B-spline functions for the space integration. After two different linearization techniques are applied, the proposed algorithms are tested on the problems of propagation of a solitary wave and interaction of two solitary waves. For the first test problem, the rate of convergence and the running time of the proposed algorithms are computed and the error norm $L_{\infty }$ is used to measure the differences between exact and numerical solutions. The three conservation quantities of the motion are calculated to determine the conservation properties of the proposed algorithms for both of the test problems.     

Extending and relating different approaches for solving fuzzy quadratic problems

Quadratic programming problems are applied in an increasing variety of practical fields. As ambiguity and vagueness are natural and ever-present in real-life situations requiring solutions, it makes perfect sense to attempt to address them using fuzzy quadratic programming problems. This work presents two methods used to solve linear problems with uncertainties in the set of constraints, which are extended in order to solve fuzzy quadratic programming problems. Also, a new quadratic parametric method is proposed and it is shown that this proposal contains all optimal solutions obtained by the extended approaches with their satisfaction levels. A few numerical examples are presented to illustrate the proposed method.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation

In this paper, a new one-dimensional space-fractional Boussinesq equation is proposed. Two novel numerical methods with a nonlocal operator (using nodal basis functions) for the space-fractional Boussinesq equation are derived. These methods are based on the finite volume and finite element methods, respectively. Finally, some numerical results using fractional Boussinesq equation with the maximally positive skewness and the maximally negative skewness are given to demonstrate the strong potential of these approaches. The novel simulation techniques provide excellent tools for practical problems. These new numerical models can be extended to two- and three-dimensional fractional space-fractional Boussinesq equations in future research where we plan to apply these new numerical models for simulating the tidal water table fluctuations in a coastal aquifer.     

一种新的声椭球无限单元

提出了一种新的声椭球无限单元．这种声无限单元基于一种新的声压表达式,这种声压表达式能够更准确地代表着椭球声场的声传播模式．这种新方法的形函数类似于Burnett方法,而权函数定义为形函数和一个附加因子的乘积．因为仅需要一维的数值积分,这种新方法的代码生成十分容易,就像处理一维单元一样．耦合标准的有限元程序,这种声无限单元理论上能够高效地求解任何形状的声源的声辐射和声散射现象．简要地推导了这种新方法,并给出了这种方法详尽的推导结果．为更有效地检验该无限元方法的可行性,文中例子仅考虑无限元求解的精度,而不包括相应的有限元．使用这种新方法,精确地推导出了摆动球的理论计算公式．而长旋转椭球的例子则表明了这种方法优于边界元方法和其他声椭球无限元方法．这些例子表明了这种新方法是切实可行的．     

基于GA-Chebyshev神经网络的分数阶Bagley-Torvik方程数值解法

本文基于现有的切比雪夫神经网络,提出了一种利用遗传算法优化切比雪夫神经网络求解分数阶Bagley-Torvik方程数值解的新方法,结合多点处的泰勒公式原理,给出数值解的一般形式,将原问题转化为求解无约束最小化问题.与现有数值方法的数值结果进行比较表明了本文方法的可行性和有效性,为分数阶微分方程中类似问题的求解提供了新的思路.     

A spectral element method for solving the Pennes bioheat transfer equation by using triangular and quadrilateral elements

A spectral element method is developed for the numerical solution of the Pennes bioheat transfer equation which models the thermal behavior of the living tissue. For the one and two dimensional cases, the implementation of this method is completely explained. In the two dimensional case, both triangular and quadrilateral elements are investigated. Through test problems, the discretization error generated from this method is reported. In the triangular elements, the error is obtained when quadrature points coincide and do not coincide with nodal points. This method is employed to solve the equation in order to obtain the temperature of the skin layers, healthy tissue, and tissue that contains the tumor.     

Approximate Solutions and Duality Theorems for Continuous-Time Linear Fractional Programming Problems

This article proposes a practical computational procedure to solve a class of continuous-time linear fractional programming problems by designing a discretized problem. Using the optimal solutions of proposed discretized problems, we construct a sequence of feasible solutions of continuous-time linear fractional programming problem and show that there exists a subsequence that converges weakly to a desired optimal solution. We also establish an estimate of the error bound. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Spectral element method for acoustic wave simulation in heterogeneous media

In this paper, we present a spectral element method for studying acoustic wave propagation in complex geological structures. Due to complexity (both lithological and stratigraphical), the use of numerical methods of higher accuracy and flexibility is needed to achieve the correct results. The spectral element method shows more accurate results compared to the low-order finite element, the conventional finite difference and the pseudospectral methods. High accuracy is reached even for rather long wave propagation times and dispersion errors are essentially eliminated; pirregular interfaces between different media can be well described so that numerical artifacts or noises are not at all introduced. The method is tested against analytical solutions both in the two-dimensional homogenous and heterogeneous media. The results of different simulations are presented.     

Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods

Automatic global error control of numerical schemes is examined. A new approach to this problem is presented. Namely, the problem is reformulated so that the global error is controlled by the numerical method itself rather than by the user. This makes it possible to find numerical solutions satisfying various accuracy requirements in a single run, which so far was considered unrealistic. On the other hand, the asymptotic equality of local and global errors, which is the basic condition of the new method for efficiently controlling the global error, leads to the concept of double quasi-consistency. This requirement cannot be satisfied within the classical families of numerical methods. However, the recently proposed peer methods include schemes with this property. There exist computational procedures based on these methods and polynomial interpolation of fairly high degree that find the numerical solution in a single run. If the integration stepsize is sufficiently small, the error of this solution does not exceed the prescribed tolerance. The theoretical conclusions of this paper are supported by the numerical results obtained for test problems with known solutions.