Variable Precision Algorithm for the Numerical Computation of the Fermi–Dirac Function ℱj(x) of Order j=−3/2

The purpose of this technical note is to present a piecewise Chebyshev expansion for the numerical computation of the Fermi–Dirac function _–3/2(x), –<x<. The variable precision algorithm we given automatically adjusts the degrees of the Chebyshev expansions so that _–3/2(x) can be efficiently computed to d significant decimal digits of accuracy, for a user specified value of d in the range 1d15.     

Convergence Analysis in the Maximum Norm of the Numerical Gradient of the Shortley–Weller Method

The Shortley–Weller method is a standard central finite-difference-method for solving the Poisson equation in irregular domains with Dirichlet boundary conditions. It is well known that the Shortley–Weller method produces second-order accurate solutions and it has been numerically observed that the solution gradients are also second-order accurate; a property known as super-convergence. The super-convergence was proved in the \(L^{2}\) norm in Yoon and Min (J Sci Comput 67(2):602–617, 2016). In this article, we present a proof for the super-convergence in the \(L^{\infty }\) norm.     

Numerical simulation of blow-up solutions for the generalized Davey–Stewartson system

Blow-up solutions for the generalized Davey–Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.     

Numerical strategies for the system of first-order IVPS using the RK–Butcher algorithm

In this article, a new method of analysis for first-order initial-value type ordinary differential equations using the Runge–Kutta (RK)–Butcher algorithm is presented. To illustrate the effectiveness of the RK–Butcher algorithm, 10 problems have been considered and compared with the RK method based on arithmetic mean, and with exact solutions of the problems, and are found to be very accurate. Stability analysis for the first-order initial-value problem (IVP) has been discussed. Error graphs for the first-order IVPs are presented in a graphical form to show the efficiency of this RK–Butcher method. This RK–Butcher algorithm can be easily implemented in a digital computer and the solution can be obtained for any length of time.     

Numerical investigation of the first bifurcation for the flow in a rotor–stator cavity of radial aspect ratio 10

The nature of transition to unsteadiness of rotor–stator disk flows of large radial aspect ratio is investigated by means of several numerical tools which consist in computing the base flow even when unstable, performing linearized or non-linear time integrations starting from initial conditions of different amplitudes and computing the spectrum of the Jacobian using the ARPACK library. From these numerical experiments we conclude that, in a cavity of radial aspect ratio 10, the transition to unsteadiness occurs through a subcritical Hopf bifurcation. In addition these calculations show the existence of a large amplitude chaotic branch for values of the Reynolds number far below the linear stability threshold, and onto which the solutions are attracted for large subcritical values due to the strong non-normality of the Jacobian of the evolution operator.     

A Study of Secure Multi-Party Elementary Function Computation Protocols

In this paper we consider three-party computations, study four operation problems of secure three-party and propose secure three-party four operations protocols. Secure three-party elementary function problems include exponential function problem, power function problem, logarithmic function problem, trigonometric function problem, and proposing a few concern protocols. The function is from four operations and compound functions, and we propose generalize method to conduct protocols. The motive of this paper is to present based on elementary function protocols for the application of secure multi-party computations.     

Numerical aspects in the finite element simulation of the Portevin–Le Chatelier effect

Serrated yielding and propagation of localized bands of plastic strain rate are the apparent phenomena of the Portevin–Le Chatelier (PLC) effect. The finite element modeling of this effect is investigated, using a model proposed by Zhang et al. [74] describing dynamic strain aging, and material parameters for a Nickel based superalloy at 500 °C. This work presents: (1) an efficient implicit integration scheme of the constitutive equations in the presence of instabilities; (2) a numerical tool to determine the type of plastic strain rate localization bands based on results of simulations; and (3) a mesh and time discretization sensitivity analysis of the model regarding the characteristics of PLC bands. This analysis is carried out in 2D and 3D for axisymmetric smooth and notched specimens.     

Numerical Methods and Simulations for the Dynamics of One-Dimensional Zakharov–Rubenchik Equations

In this paper, we propose and study several accurate numerical methods for solving the one-dimensional Zakharov–Rubenchik equations (ZRE). We begin with a review on the important properties of the ZRE, including the solitary wave solutions and the various conservation laws. Then we propose a very efficient and accurate numerical method based on the time-splitting technique and the Fourier pseudo-spectral (TSFP) method. Next, we propose some conservative and non-conservative types of finite difference time domain methods, including a Crank–Nicolson finite difference method that conserves the mass and the energy of the system in the discrete level. Discrete conservation laws and numerical stability of all the proposed methods are analyzed. Comparisons between different methods in the efficiency, stability and accuracy are carried out, which identifies that the TSFP method is the most efficient and accurate numerical method among all the methods. Lastly, we apply the TSFP method to simulate and study the dynamics of the solitons in the ZRE numerically.     

Computation of the Interval of Stability of Runge–Kutta–Nyström Methods

We present a Mathematica package to compute the interval of stability of Runge–Kutta–Nystrom methods fory">=f(t,y).     

Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System

In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn–Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We solve this issue by developing two linear and decoupled, first order and a second order time-stepping schemes using the so-called “invariant energy quadratization” approach for the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling potential. Moreover, the resulting linear system is well-posed and the linear operator is symmetric positive definite. We rigorously prove the first order scheme is unconditionally energy stable. Various numerical simulations are presented to demonstrate the stability and the accuracy thereafter.     

Hermite–Birkhoff Interpolation Polynomial of Minimum Norm in Hilbert Space

Cybernetics and Systems Analysis - The Hermite–Birkhoff interpolation problem for a nonlinear operator in the Hilbert space is considered. For this problem, the theorem on the interpolation...     

Call for Papers for the 3rd Annual Conference of Theory and Applications of Models of Computation

May 15th to 20th, 2006, Beijing, China http://gcl.iscas.ac.cn/accl06/TAMC06_Home.htmTAMC 2006 is a new annual conference focusing on theory and applications of computation. It is organized as part of the Grand China NSF International Joint Project after which the conference is named, and is supported by the Chinese National Science Foundation, and the Institute of Software, the Chinese Academy of Sciences. Previously two annual meetings were held in 2004 and 2005, with enthusiastic …     

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Concise and explicit formulas for dyadic Green’s functions, representing the electric and magnetic fields due to a dipole source placed in layered media, are derived in this paper. First, the electric and magnetic fields in the spectral domain for the half space are expressed using Fresnel reflection and transmission coefficients. Each component of electric field in the spectral domain constitutes the spectral Green’s function in layered media. The Green’s function in the spatial domain is then recovered involving Sommerfeld integrals for each component in the spectral domain. By using Bessel identities, the number of Sommerfeld integrals are reduced, resulting in much simpler and more efficient formulas for numerical implementation compared with previous results. This approach is extended to the three-layer Green’s function. In addition, the singular part of the Green’s function is naturally separated out so that integral equation methods developed for free space Green’s functions can be used with minimal modification. Numerical results are included to show efficiency and accuracy of the derived formulas.     

Numerical appraisal of three low Mach number algorithms for radiative–convective flows in enclosures

The present study investigates three different algorithms for the numerical simulation of non-Boussinesq convection with thermal radiative heat transfer based on a low-Mach number formulation. The solution methodology employs a fractional step approach based on the finite-volume method on arbitrary polyhedral meshes. The three algorithms compute the coupled governing equations in a segregated manner using the conservative form of momentum equations in conjunction with a variable coefficient pressure Poisson equation. The first algorithm (Algorithm A) uses conservation of mass and energy equation to compute density and temperature. The other two algorithms (Algorithm B) and (Algorithm C) calculates temperature and density from the equation of state respectively and solves a conservative form of the continuity and energy equation to obtain density and temperature respectively. The energy and mass conservation errors arising due to the use of Algorithms B and C are derived concerning various non-dimensional parameters governing the flow and heat transfer. The significance of these errors is highlighted by performing investigations over a range of Rayleigh, Prandtl, and Planck numbers for various two and three-dimensional natural convection problems with radiative heat transfer. Finally, the role of balancing of the pressure and buoyancy terms is emphasized for robust calculations of large temperature difference thermo-buoyant convection with radiative heat transfer.     

Numerical exploration of vortex matter in Bose–Einstein condensates

We describe a finite element numerical approach to the full Hartree-Fock-Bogoliubov treatment of a vortex lattice in a rapidly rotating Bose–Einstein condensate. We study the system in the regime of high thermal or significant quantum fluctuations where we are presented with a very large nonlinear unsymmetric eigenvalue problem which is indefinite and which possesses low-lying excitations clustered arbitrarily close to zero, a problem that requires state-of-the-art numerical techniques.     

A Numerical Method to Solve Identification Problem for the Lower Coefficient and the Source in the Convection–Reaction Equation

We consider the problem of identifying simultaneously the kinetic reaction coefficient and source function depending only on a spatial variable in one-dimensional linear convection–reaction equation. As additional conditions, a non-local integral condition for the solution of the equation and condition of final overdetermination are given. This problem belongs to the class of combined inverse problems. By integrating the equation with the use of additional integral condition, the problem is transformed to a coefficient inverse problem with local conditions. The derivative with respect to the spatial variable is discretized and a special representation is proposed to solve the resultant semi-discrete problem. As a result, for each discrete value of the spatial variable, the semi-discrete problem splits into two parts: a Cauchy problem and a linear equation with respect to the approximate value of the unknown kinetic coefficient. To determine the source function, an explicit formula is also obtained. The numerical solution of the Cauchy problem uses the implicit Euler method. Numerical experiments are carried out on the basis of the proposed method.     

Super-Convergence in Maximum Norm of the Gradient for the Shortley–Weller Method

We prove in this paper the second-order super-convergence in \(L^{\infty }\)-norm of the gradient for the Shortley–Weller method. Indeed, this method is known to be second-order accurate for the solution itself and for the discrete gradient, although its consistency error near the boundary is only first-order. We present a proof in the finite-difference spirit, using a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. The proof is based on a discrete Poisson equation for the discrete gradient, with second-order accurate Dirichlet boundary conditions. The advantage of this finite-difference approach is that it can provide pointwise convergence results depending on the local consistency error and the location on the computational domain.     

A Least-Squares/Relaxation Method for the Numerical Solution of the Three-Dimensional Elliptic Monge–Ampère Equation

In this article, we address the numerical solution of the Dirichlet problem for the three-dimensional elliptic Monge–Ampère equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities. Dedicated numerical solvers are derived for the efficient solution of the local optimization problems with cubicly nonlinear equality constraints. The approximation relies on mixed low order finite element methods with regularization techniques. The results of numerical experiments show the convergence of our relaxation method to a convex classical solution if such a solution exists; otherwise they show convergence to a generalized solution in a least-squares sense. These results show also the robustness of our methodology and its ability at handling curved boundaries and non-convex domains.     

Numerical efficiency of some exponential methods for an advection–diffusion equation

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them.     

Numerical solution of the fractional Rayleigh–Stokes model arising in a heated generalized second-grade fluid

This paper addresses the solution of the Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid using fractional derivatives and the radial basis function-generated finite difference (RBF-FD) method. The time discretization is accomplished via the finite difference approach, while the spatial derivative terms are discretized using the local RBF-FD. The main idea is to consider the distribution of the data nodes within the local support domain so that the number of nodes remains constant. In addition, the stability and convergence analysis of the proposed method are discussed. The results using the RBF-FD are compared with those of other techniques on irregular domains showing the feasibility and efficiency of the new approach.