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 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.     

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 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 schemes for the forward–backward heat equation

This paper is concerned with a class of forward–backward heat equations. We use Saulyev's scheme to formulate certain approximation schemes. Then a non-overlap domain decomposition method is presented for the numerical solution. The numerical experiments show that the given algorithm is feasible and effective.     

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.     

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.     

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.     

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).     

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...     

Numerical minimization¶of the Mumford–Shah functional

We consider a discrete approximation of the Mumford–Shah functional defined on finite element spaces. For a relevant model case we study global and local minima; from their behavior, we are led to propose two minimization methods, based on the quasi-Newton algorithm, which can find the absolute minimum of the functional. Received: January 2000 / Accepted: October 2000     

Computation of sub–optimal linear controls for non–linear stochastic systems†

A successive approximation technique based on statistical linearization is developed which permits the design of sub–optimal linear controls for non–linear time–invariant processes perturbed by Gaussian random disturbances. The application of the technique to a chemical process example is given, and the validity of the results is confirmed by simulation.     

Computation of the padé table

Simple recursion formulas are presented for computing the coefficients in the Padé table of a power series f(x). A Fortran subroutine for the computation is given in an Appendix, where a second subroutine for evaluating the elements for any x is also given. Two applications are mentioned. The assumption is made that the Padé table is normal.     

Computation of the state transition matrix for a class of delay-differential systems†

Control systems can drift into instability or, less catastrophically, exhibit resonance behaviour. One role for adaptive controllers is to learn sufficient information concerning the dominant closed-loop system mode so as to apply effective feedback to dampen these modes. In such situations the adaptive loop augments the fixed controller feedback loop. This paper presents an algorithm for adaptive resonance suppression and provides simulation results to study its behaviour in the presence of high-order unmodelled dynamics. The algorithm appears particularly useful for enhancing existing fixed controller designs.     

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.