An efficient numerical method for simulating the dynamics of coupling Bose–Einstein condensates in optical resonators

In this paper, we present an efficient numerical method for computing the dynamics of coupling Bose–Einstein condensates in optical resonators at extremely low temperature, which is modeled by Gross–Pitaevskii equations (GPEs) coupled with an integral and ordinary differential equation (IODE). Our numerical method is based on an integration factor method for solving the IODE and a time-splitting sine pseudospectral method for solving the GPEs. Our numerical method keeps well the dynamical properties of the mathematical model and have spectral accuracy in space. Through extensive numerical simulations, we analyze which factors may be useful for uniting Bose–Einstein condensates in optical resonators and study the possible way of dynamically uniting two Bose–Einstein condensates in optical resonators.     

Modal Hermite spectral collocation method for solving multi-dimensional hyperbolic telegraph equations

The present research is contemplated proposing a numerical solution of multi-dimensional hyperbolic telegraph equations with appropriate initial time and boundary space conditions. The truncated Hermite series with unknown coefficients are used for approximating the solution in both of the spatial and temporal variables. The basic idea for discretizing the considered one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) telegraph equations is based on the collocation method together with the Hermite operational matrices of derivatives. The resulted systems of linear algebraic equations are solved by some efficient methods such as LU factorization. The solution of the algebraic system contains the coefficients of the truncated Hermite series. Numerical experiments are provided to illustrate the accuracy and efficiency of the presented numerical scheme. Comparisons of numerical results associated to the proposed method with some of the existing numerical methods confirm that the method is accurate and fast experimentally.     

A multi-domain Legendre spectral collocation method for nonlinear neutral equations with piecewise continuous argument

In this paper, for the neutral equations with piecewise continuous argument, we construct a spectral collocation method by combining the shifted Legendre–Gauss–Radau interpolation and a multi-domain division. Based on the non-classical Lipschitz condition, the convergence results of the method are derived. The results show that the method can arrive at high accuracy under the suitable conditions. Several numerical examples further illustrate the obtained theoretical results and the computational effectiveness of the method.     

A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

This paper presents a sparse collocation method for solving the time-dependent Hamilton–Jacobi–Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite time-horizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the time-dependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation.     

Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation

In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary conditions. Based on the autocorrelation function of Daubechies scaling functions, collocation method is conducted for the spatial discretization. The obtained semi-discrete system is proved to have semi-discrete multi-symplectic conservation laws and semi-discrete energy conservation laws. Then, appropriate symplectic scheme is applied for time integration, which leads to full-discrete multi-symplectic conservation laws. Numerical experiments for the nonlinear Schrödinger equation and Camassa–Holm equation show the high accuracy, effectiveness and good conservation properties of the proposed method.     

A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ. A stability analysis is performed, showing that even for the explicit scheme that θ=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.     

Optimality of an error estimate of a one-step numerical integration method for certain initial value problems

In a recent paper, an error estimate of a one-step numerical method, originated from the Lanczos tau method, for initial value problems for first order linear ordinary differential equations with polynomial coefficients, was obtained, based on the error of the Lanczos econo-mization process. Numerical results then revealed that the estimate gives, correctly, the order of the tau approximant being sought. In the present paper we further establish that the error estimate is optimum with respect to the integration of the error equation. Numerical examples are included for completeness.     

Development of a mixed control volume – Finite element method for the advection–diffusion equation with spectral convergence

In this paper we attack the problem of devising a finite volume method for computational fluid dynamics and related phenomena which can deal with complex geometries while attaining high-orders of accuracy and spectral convergence at a reasonable computational cost. As a first step towards this end, we propose a control volume finite element method for the solution of the advection–diffusion equation. The numerical method and its implementation are carefully tested in the paper where h- and p-convergence are checked by comparing numerical results against analytical solutions in several relevant test-cases. The numerical efficiency of a selected set of operations implemented is estimated by operation counts, ill-conditioning of coefficient matrices is avoided by using an appropriate distribution of interpolation points and control-volume edges.     

A direct spectral domain decomposition method for the computation of rotating flows in a T-shape geometry

This paper presents a direct domain decomposition method, coupled with a Chebyshev collocation approximation, for solving the incompressible Navier-Stokes equations in the vorticity-streamfunction formulation. The method is based on the influence matrix technique used to treat the lack of vorticity boundary conditions on no-slip walls as well as to enforce the continuity conditions at the interfaces between adjacent subdomains. The multi-domain approach is proposed in order to extend the use of spectral approximations to non-rectangular geometries and singular solutions. It is applied to the computation of a four domain configuration, corresponding to a forced throughflow in a rotating channel-cavity system which is important in air cooling devices and cannot be modeled by single-domain spectral approximations.     

A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation

This paper presents a numerical solution for solving a nonlinear 2-D optimal control problem (2DOP). The performance index of a nonlinear 2DOP is described with a state and a control function. Furthermore, dynamic constraint of the system is given by a classical diffusion equation. It is preferred to use the Ritz method for finding the numerical solution of the problem. The method is based upon the Legendre polynomial basis. By using this method, the given optimisation nonlinear 2DOP reduces to the problem of solving a system of algebraic equations. The benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, compared with the eigenfunction method, the satisfactory results are obtained only in a small number of polynomials order. This numerical approach is applicable and effective for such a kind of nonlinear 2DOP. The convergence of the method is extensively discussed and finally two illustrative examples are included to observe the validity and applicability of the new technique developed in the current work.     

Spectral and Haar wavelet collocation method for the solution of heat generation and viscous dissipation in micro-polar nanofluid for MHD stagnation point flow

The aim and significance of paper presents, the semi-numerical investigation of magnetohydrodynamic flow of micropolar nanofluid with stagnation point is carried out under the influence of viscous dissipation and heat generation. The micropolar nanofluids are electrically conducting non-Newtonian fluids. The important applications of these fluids are observed in many research areas viz. bioengineering, biofuels and biomedical sectors etc. The appropriate similarity transformations are used to transform the governing equations into system of coupled nonlinear ordinary differential equations and are solved by using shifted Chebyshev collocation method and Haar wavelet collocation method. The variations in velocity, angular velocity, temperature and concentration profiles under the impact of various physical parameters, characterizing the flow field are discussed and are presented via graphs and tables. Temperature enhancement occurs with increment in each parameter except for Prandtl number. The concentration near the surface decreases with increment in the values of parameters and gradually it increases, except for Prandtl number and Schmidt number. The reverse trend of heat transfer occurs​ near a surface, when the dominance of stream velocity over stretching velocity is observed.     

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted $L^2$- and $L^{\infty }$-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order $min\{2?\alpha ,1+\alpha \}$, where $\alpha \in (0,1)$ is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.     

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.     

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.     

A regularization method for a Cauchy problem of Laplace's equation in an annular domain

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace's equation in an annular domain. Based on a conditional stability for the Cauchy problem of Laplace's equation, we obtain a convergence estimate under the suitable choice of a regularization parameter and an a-priori bound assumption on the solution. A numerical example is provided to show the effectiveness of the proposed method from both accuracy and stability.     

Flatness-based trajectory planning for diffusion–reaction systems in a parallelepipedon—A spectral approach

Spectral analysis is considered for the flatness-based solution of the trajectory planning problem for a boundary controlled diffusion–reaction system defined on a 1≤r-dimensional parallelepipedon. By exploiting the Riesz spectral properties of the system operator, it is shown that a suitable reformulation of the resolvent operator allows a systematic introduction of a basic output, which yields a parametrization of both the system state and the boundary input in terms of differential operators of infinite order. Their convergence is verified for both infinite-dimensional and finite-dimensional actuator configurations by restricting the basic output to certain Gevrey classes involving non-analytic functions. With this, a systematic approach is introduced for basic output trajectory assignment and feedforward tracking control towards the realization of finite-time transitions between stationary profiles.     

A semi-local convergence theorem for a robust revised Newton’s method

It is well known that Newton’s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newton’s method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton–Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newton’s method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newton’s method.     

Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method

In this paper, we propose a regular perturbation method to obtain approximate analytic solutions of exterior and interior Dirichlet problems for Laplace’s equation in planar domains. This method, starting from a geometrical perturbation of these planar domains, reduces our problems to a family of classical Dirichlet problems for Laplace’s equation in a circle. Numerical examples are given and comparisons are made with the solutions obtained by other approximation methods.     

Development of a stakeholder identification and analysis method for human factors integration in work system design interventions – Change Agent Infrastructure

In any work system design intervention—for example, a physical workplace re-design, a work process change, or an equipment upgrade—it is often emphasized how important it is to involve stakeholders in the process of analysis and design, to gain their perspectives as input to the development, and ensure their future acceptance of the solution. While the users of an artifact or workplace are most often regarded as being the most important stakeholders in a design intervention, in a work-system context there may be additional influential stakeholders who influence and negotiate the design intervention's outcomes, resource allocation, requirements, and implementation. Literature shows that it is uncommon for empirical ergonomics and human factors (EHF) research to apply and report the use of any structured stakeholder identification method at all, leading to ad-hoc selections of whom to consider important. Conversely, other research fields offer a plethora of stakeholder identification and analysis methods, few of which seem to have been adopted in the EHF context. This article presents the development of a structured method for identification, classification, and qualitative analysis of stakeholders in EHF-related work system design intervention. It describes the method's EHF-related theoretical underpinnings, lessons learned from four use cases, and the incremental development of the method that has resulted in the current method procedure and visualization aids. The method, called Change Agent Infrastructure (abbreviated CHAI), has a mainly macroergonomic purpose, set on increasing the understanding of sociotechnical interactions that create the conditions for work system design intervention, and facilitating participative efforts.