Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.     

Performance and numerical behavior of the second‐order scheme of precise time‐step integration for transient dynamic analysis

Spurious high‐frequency responses resulting from spatial discretization in time‐step algorithms for structural dynamic analysis have long been an issue of concern in the framework of traditional finite difference methods. Such algorithms should be not only numerically dissipative in a controllable manner, but also unconditionally stable so that the time‐step size can be governed solely by the accuracy requirement. In this article, the issue is considered in the framework of the second‐order scheme of the precise integration method (PIM). Taking the Newmark‐β method as a reference, the performance and numerical behavior of the second‐order PIM for elasto‐dynamic impact‐response problems are studied in detail. In this analysis, the differential quadrature method is used for spatial discretization. The effects of spatial discretization, numerical damping, and time step on solution accuracy are explored by analyzing longitudinal vibrations of a shock‐excited rod with rectangular, half‐triangular, and Heaviside step impact. Both the analysis and numerical tests show that under the framework of the PIM, the spatial discretization used here can provide a reasonable number of model types for any given error tolerance. In the analysis of dynamic response, an appropriate spatial discretization scheme for a given structure is usually required in order to obtain an accurate and meaningful numerical solution, especially for describing the fine details of traction responses with sharp changes. Under the framework of the PIM, the numerical damping that is often required in traditional integration schemes is found to be unnecessary, and there is no restriction on the size of time steps, because the PIM can usually produce results with machine‐like precision and is an unconditionally stable explicit method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

A numerical scheme to the McKendrick–von Foerster equation with diffusion in age

In this paper a numerical scheme for McKendrick–von Foerster equation with diffusion in age (MV‐D) is proposed. First, we discretize the time variable to get a second‐order ordinary differential equation (ODE). At each time level, well‐posedness of this ODE is established using classical methods. Stability estimates for this semidiscrete scheme are derived. Later we construct piecewise linear (in time) functions using the solutions of the semidiscrete problems to approximate the solution to MV‐D and establish the convergence result. Numerical results are presented in some cases and compared with the corresponding analytic solutions where the latter is known explicitly.     

A local radial basis function collocation method to solve the variable‐order time fractional diffusion equation in a two‐dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable‐order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real‐world diffusion is usually non‐Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two‐dimensional, variable‐order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill‐conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root‐mean‐square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable‐order TFDE in two‐dimensional irregular domains.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann‐Liouville derivative

This article deals with the design, analysis, and implementation of a robust numerical scheme when applied to time‐fractional reaction‐diffusion system. Stability analysis and numerical treatment of chaotic fractional differential system in Riemann‐Liouville sense are considered in this article. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise. Numerical results presented agree with the theoretical findings.     

Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition

In this work the combined finite difference and spectral methods have been proposed for the numerical solution of the one‐dimensional wave equation with an integral condition. The time variable is approximated using a finite difference scheme. But the spectral method is employed for discretizing the space variable. The main idea behind this approach is that we can get high‐order results. The new method is used for two test problems and the numerical results are obtained to support our theoretical expectations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Approximating the inverse and the Moore‐Penrose inverse of complex matrices

A parametric family of fourth‐order schemes for computing the inverse and the Moore‐Penrose inverse of a complex matrix is designed. A particular value of the parameter allows us to obtain a fifth‐order method. Convergence analysis of the different methods is studied. Every iteration of the proposed schemes involves four matrix multiplications. A numerical comparison with other known methods, in terms of the average number of matrix multiplications and the mean of CPU time, is presented.     

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017     

Efficient structure‐preserving schemes for good Boussinesq equation

The good Boussinesq equation is endowed with symplectic conservation law and energy conservation law. In this paper, some new highly efficient structure‐preserving methods for the good Boussinesq equation are proposed by improving the standard finite difference method (FDM). The new methods only use and calculate values at the odd (or even) nodes to reduce the computational cost. We call this kind of methods odd‐even method (OEM). Numerical results show that the OEM and the standard FDM have nearly the same numerical errors under the same mesh partition. However, the OEM is much more efficient than the standard FDM, such as the consumed CPU time and occupied memory.     

Multilevel numerical solutions of convection‐dominated diffusion problems by spline wavelets

In this article, we utilize spline wavelets to establish an adaptive multilevel numerical scheme for time‐dependent convection‐dominated diffusion problems within the frameworks of Galerkin formulation and Eulerian‐Lagrangian localized adjoint methods (ELLAM). In particular, we shall use linear Chui‐Quak semi‐orthogonal wavelets, which have explicit expressions and compact supports. Therefore, both the diffusion term and boundary conditions in the convection‐diffusion problems can be readily handled. Strategies for efficiently implementing the scheme are discussed and numerical results are interpreted from the viewpoint of nonlinear approximation. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

Finite‐region stability and finite‐region boundedness for 2D Roesser models

In this paper, we establish finite‐region stability (FRS) and finite‐region boundedness analysis methods to investigate the transient behavior of discrete two‐dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite‐region stabilization issue is analyzed for the corresponding two‐dimensional closed‐loop system. Similar to FRS analysis, the finite‐region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Perturbed fixed point iterative methods based on pattern structure

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi‐nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string‐averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations. We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography.     

Extension of high‐order compact schemes to time‐dependent problems

In this article, we present an extension of our previous approaches for steady‐state higher‐order compact (HOC) difference methods to time‐dependent problems. The formulation also provides a framework for similar treatment of other HOC spatial schemes. A stability analysis is provided for transient convection‐diffusion in 1D and transient diffusion in 2D. Supporting numerical experiments are included to illustrate stability and accuracy as well as oscillatory and dissipative behavior. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 657–672, 2001     

Numerical approximation of Quasi‐Newtonian flows by ALE‐FEM

The aim of this work is to investigate the numerical approximation of a nonlinear, time‐dependent quasi‐Newtonian flow problem formulated in the framework of Arbitrary Lagrangian Eulerian method. We present some stability results and convergence analysis of finite element solutions for semidiscrete and fully discretized problems, respectively. Numerical results supporting the derived error estimate are also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012