区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.     

A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

A new shift‐adaptive meshfree method for solving a class of time‐dependent partial differential equations (PDEs) in a bounded domain (one‐dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid‐free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long‐wave equation and a Korteweg‐de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622–1646, 2016     

Application of the Galerkin method to the formulation of a three-dimensional nonlinear hydrodynamic numerical sea model

A mathematical formulation is presented for solving the three-dimensional nonlinear hydrodynamic equations, using the Galerkin method with an arbitrary set of basis functions.An explicit time splitting method is used to integrate these equations through time. The time splitting method is formulated in such a way that the advective terms, which are computationally expensive to evaluate, are integrated with a longer time step than the linear terms. The length of the time step used to integrate the linear terms is determined by the propagation speed of the gravity waves. The paper demonstrates that using this time splitting method an accurate and computationally economic solution of the full three-dimensional equations is possible.Numerical results are presented for the nonlinear seiche motion in a one-dimensional basin, and for the three-dimensional wind induced flow in a closed rectangular basin, using basis sets of cosine functions, Chebyshev polynomials and Gram-Schmidt orthogonalized polynomials.     

Complex diffusion filtering of three dimensional turbulent flows

The linear and nonlinear complex diffusion filtering methods are proposed to extract the organized coherent part as well as the random incoherent part from forced and decaying turbulent flows. An attempt to examine the robustness of the two methods in filtering the turbulent flow field without the transformation into the frequency domain is carried out. The velocity fields of the forced and decaying cases are decomposed into coherent and incoherent parts in the spatial domain. The complex diffusion process can be obtained by combining the linear diffusion equation and the free particle Schrodinger equation. The imaginary parts in the two methods serve as a robust edge-detector with increasing confidence in time. The numerical implementations of the 3D linear and nonlinear complex diffusion partial differential equations are done using the finite difference method. The flatness, skewness and spectrum of the extracted fields are also studied for each filtering method. Results show that the two filtering methods can identify the coherent fields and preserve the features of the turbulent fields. Comparisons to the wavelet and Fourier decompositions are also considered.     

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.     

Numerical Analysis of a High-Order Scheme for Nonlinear Fractional Differential Equations with Uniform Accuracy

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on each subinterval. The method is shown to be unconditionally stable, and for general nonlinear equations, the uniform sharp numerical order 3 − $ν$ can be rigorously proven for sufficiently smooth solutions at all time steps. The proof provides a general guide for proving the sharp order for higher-order schemes in the nonlinear case. Some numerical examples are given to validate our theoretical results.     

Numerical comparison of three stochastic methods for nonlinear PN junction problems

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.     

An advanced study on the solution of nanofluid flow problems via Adomian’s method

Nanofluid flow is one of the most important areas of research at the present time due to its wide and significant applications in industry and several scientific fields. The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with boundary conditions at infinity. These boundary conditions at infinity cause difficulties for any of the series method, such as Adomian’s method, the variational iteration method and others.The objective of the present work is to introduce a reliable method to overcome such difficulties that arise due to an infinite domain. The proposed scheme, that we will introduce, is based on Adomian’s decomposition method, where we will solve a system of nonlinear differential equations describing the boundary layer flow of a nanofluid past a stretching sheet.     

基于Euler方程有限差分方法驻波晃动模拟

研究在二维水槽带非线性自由面边界条件的Euler方程的数值解,数值模拟了驻波的波高．将不规则的物理区域变换为一个固定的正方形计算区域,在计算区域使用交错网格技术的目的是准确捕捉流场瞬间的波高值,应用由Bang-fuh Chen建立的时间无关有限差分方法求解不可压缩无粘Euler方程的数值解．通过数值结果表明,数值解很好地吻合分析解和以前出版的文献结果．从数值解可以看出,非线性现象和拍的现象非常明显,同时数值模拟了带初始驻波的水平激励和垂直激励运动,具有很好的数值效果．     

A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES

In this work, we propose an efficient numerical method for computing the electrostaticinteraction between two like-charged spherical particles which is governed by the nonlinearPoisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterativemethod which leads to a sequence of linearized equations. A modified central finite differ-ence scheme is developed to solve the linearized equations on an exterior irregular domainusing a uniform Cartesian grid. With uniform grids, the method is simple, and as aconsequence, multigrid solvers can be employed to speed up the convergence. Numericalexperiments on cases with two isolated spheres and two spheres confined in a chargedcylindrical pore are carried out using the proposed method. Our numerical schemes arefound efficient and the numerical results are found in good agreement with the previouspublished results.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.     

Chebyshev tau meshless method based on the highest derivative for fourth order equations

It is well known that the numerical integration process is much less sensitive than numerical differential process when dealing with the differential equations. After integration, accuracy is no longer limited by that of the slowly convergent series for the highest derivative, but only by that of the unknown function itself. In this paper, a Chebyshev tau meshless method based on the highest derivative (CTMMHD) is developed for fourth order equations on irregularly shaped domains with complex boundary conditions. The problem domain is embedded in a domain of regular shape. The integration and multiplication of Chebyshev expansions are given in matrix representations. Several numerical experiments including standard biharmonic problems, problems with variable coefficients and nonlinear problems are implemented to verify the high accuracy and efficiency of our method.     

An SQP approach with line search for a system of nonlinear equations

In this paper, a line search sequential quadratic programming (SQP) approach to a system of nonlinear equations (SNE) is taken. In this method, the system of nonlinear equations is transformed into a constrained nonlinear programming problem at each step, which is then solved using SQP algorithms with a line search strategy. Furthermore, at each step, some equations, which are satisfied at the current point, are treated as constraints and the others act as objective functions. In essence, constrained optimization strategies are utilized to cope with the system of nonlinear equations.     

Polygons of differential equations for finding exact solutions

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg–de Vries–Burgers equation, the generalized Kuramoto–Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order Korteveg–de Vries equation, the fifth-order modified Korteveg–de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.     

Numerical solution of nonlinear diffraction problems

An adaptive finite element method is developed for solving Maxwell's equations in a nonlinear periodic structure. The medium or computational domain is truncated by a perfect matched layer (PML) technique. Error estimates are established. Numerical examples are provided, which illustrate the efficiency of the method.     

Analysis of the effects of a pulsed electromagnetic field on the dynamic response of electrically conductive composites

The effects of pulsed electromagnetic fields on the dynamic mechanical response of electrically conductive anisotropic plates are studied. The analysis is based on the simultaneous solving of the system of nonlinear partial differential equations that include equations of motion and Maxwell’s equations. Physics-based hypotheses for electro-magneto-mechanical coupling in anisotropic composite plates and dimension reduction solution procedures for the nonlinear system of the governing equations are presented. A numerical solution procedure for the resulting two-dimensional nonlinear system of the governing equations has been developed and consists of the sequential application of time and spatial integration and quasilinearization. The developed methodology is applied to the problem of the dynamic response of a long current-carrying unidirectional carbon fiber polymer matrix composite plate subjected to transverse impact load and in-plane pulsed electromagnetic load. The interacting effects of the pulsed electric current, external magnetic field, and mechanical load are studied.