Efficient Finite Difference Research Articles

Heat transfer in MHD stagnation point flow of a ferrofluid over a stretchable rotating disk

In this article, heat transfer analysis in three dimensional boundary layer stagnation point flow of incompressible ferrofluid over a stretchable rotating circular disk in the presence of uniform external magnetic field has been investigated. For this purpose, three different types of ferroparticles namely, magnetite (Fe3O4), cobalt ferrite (CoFe2O4) and Mn-Zn ferrite (Mn-ZnFe2O4) are considered with water as a base fluid. The governing partial differential equations of the considered problem are transformed into non-dimensional ordinary differential equations using similarity transformation (Turkyilmazoglu, 2012). The solution of obtained equations is computed numerically using an efficient implicit finite difference scheme. The effects of pertinent parameters namely, volume fraction parameter of magnetic nanoparticle, rotating parameter, magnetic parameter, velocity ratio parameter and Prandtl number on both radial and azimuthal velocity profiles, temperature profile, skin friction coefficients and local Nusselt number are computed and shown through graphs and tables. For magnetite ferroparticle (Fe3O4) radial velocity decreases and azimuthal velocity and temperature increase with increasing value of volume fraction parameter ϕ. The values of skin friction coefficient in radial direction and local Nusselt number of magnetite ferroparticle (Fe3O4) become higher than that of other ferroparticles due to the maximum density and thermal conductivity.

Efficient Numerical Methods for Solving Large Henri-Michaelis-Menten Systems

Systems of nonlinear differential equations arise in the mathematical modelling of enzyme-substrate dynamics, see, for example, [1], [2] and [3]. It is necessary to solve these systems numerically, which may be difficult when the system is large. These systems usually exhibit singular perturbations, which add further difficulties to the computation of their numerical solutions. Here, efficient finite difference methods are discussed, which use specially constructed piecewise uniform meshes and appropriate standard finite difference operators to overcome these difficulties. ...

Influence of higher order chemical reaction and non-uniform heat source/sink on Casson fluid flow over a vertical cone and flat plate

The present study focuses on the effects of surface dependent heat source/sink, viscosity and thermal conductivity variations on unsteady flow of Casson fluid over a vertical cone and flat plate with the influence of thermal radiation, temperature dependent heat source/sink and higher order chemical reaction effects. The model which is constituted with highly nonlinear governing equations is solved numerically by an efficient finite difference scheme of Crank–Nicolson type. Numerical calculations are carried out and reported graphically for various parametric conditions to show interesting aspects of the solution on flow, heat and mass transfer characteristics of this problem. Results indicate that the hydrodynamic and thermal boundary layer thickness decrease for higher values of thermal radiation parameter. The fluid flow and heat transfer are appreciably influenced by the temperature dependent heat source/sink parameter. The flow and heat transfer characteristics of this problem are also regulated by surface dependent heat source/sink, viscosity and thermal conductivity variation parameters.

Transient Combined Convective Heat Transfer over a Stretching Surface in a Non-Newtonian Nanofluid Using Buongiorno’s Model

The present paper investigates the transient mixed convective boundary layer flow of an incompressible non-Newtonian quiescent nanofluid adjacent to a vertical stretching surface. The effects of the Brownian motion and thermophoresis are included for the nanofluid. Using appropriate non-similarity transformations the non-dimensional, coupled and highly non-linear system of equations is solved numerically using the efficient Keller-box implicit finite difference method for the whole transient from t=0 (initial state) to (final steady-state flow). The box method is unconditionally stable. Numerical results for dimensionless velocity (f’), micro-rotation (g), temperature (θ), nanoparticle volume fraction (Φ) at final steady state flow, skin friction function (), Nusselt number function () and Sherwood number function () have been presented on various parameters inform of tables and graphs. The results indicate that as Nb and Nt increase, the Nusselt number decreases whereas Sherwood number increases at initial and early state time but decreases at the final steady state time. As the K increases, the friction factor decreases whereas surface mass transfer rate and the surface heat transfer rates slightly increase. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work. The study has many practical applications such as extrusion of plastic sheets, paper production, glass blowing, metal spinning and drawing plastic films.

Mixed convection flow over a horizontal circular cylinder in a viscous fluid at the lower stagnation point with convective boundary conditions

In this paper, we study the steady mixed convection boundary layer flow over a horizontal circular cylinder in a viscous fluid with the bottom surface of the cylinder heated by convection from a hot fluid. The resulting system of nonlinear partial differential equations is solved numerically using an efficient implicit finite-difference scheme known as the Keller box method along with Newton’s linearization techniques. The special case at the lower stagnation point of the cylinder is considered. The solutions for the flow and heat transfer characteristics are obtained and presented for various values of the governing parameters, namely, the mixed convection parameter, the Prandtl number, and the conjugate parameter. The effects of these parameters on the flow and heat transfer characteristic are thoroughly examined in this study.

On the multi-symplectic structure of the Serre–Green–Naghdi equations

In this short note, we present a multi-symplectic structure of the Serre–Green–Naghdi (SGN) equations modelling nonlinear long surface waves in shallow water. This multi-symplectic structure allow the use of efficient finite difference or pseudo-spectral numerical schemes preserving exactly the multi-symplectic form at the discrete level.

Three-dimensional stagnation flow of a nanofluid containing both nanoparticles and microorganisms on a moving surface with anisotropic slip

In this paper, we have extended Wang’s (2013) problem of the three-dimensional stagnation flow of a Newtonian fluid on a moving plate with anistropic slip to the case of a nanofluid in suspension of both the nanoparticles and microorganisms. The passively controlled nanofluid model is taken into account to describe this problem, which is believed to be physically more realistic than previously commonly used models. The problem is then reduced by a set of similarity transformations to seven coupled nonlinear ordinary equations with coupled boundary conditions. Numerical solutions are given by means of a very efficient finite difference method. The influences of various physical parameters on the distributions of the velocity, the temperature, the nanoparticle volumetric fraction, the density of motile microorganisms profiles, as well as the local skin friction coefficient, the local Nusselt number, the local wall mass flux and the local density of the motile microorganisms are presented and discussed.

Vibration analysis of Euler–Bernoulli nanobeams embedded in an elastic medium by a sixth-order compact finite difference method

This paper presents an efficient sixth-order finite difference discretization for vibration analysis of a nonlocal Euler–Bernoulli beam embedded in an elastic medium. The Pasternak elastic foundation model is utilized to represent the surrounding elastic medium. Nonlocal differential elasticity of Eringen is exploited to reveal the nun-locality effect of nanobeams. Sixth-order accuracy schemes are developed for discretization of both the governing equation and boundary conditions. Sixth-order accuracy schemes are derived for simply supported, clamped and free boundary conditions. Numerical results include comparison with exact solutions and with previously published works are presented for the fundamental frequencies. In addition, numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, and boundary conditions on the dynamic characteristics of the beam.

An efficient finite-difference scheme for computation of electron states in free-standing and core–shell quantum wires

The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schrödinger equation, which is written in cylindrical coordinates φ, ρ, and z. We show that a direct discretization of the Schrödinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea et al. (2008), which replaces the wave function ψ(ρ) with the function F(ρ)=ψ(ρ)ρ and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schrödinger equation in the first step, and then applies the Liouville-like transformation to the difference equation. Such a procedure gives a symmetric Hamiltonian matrix, resulting in an accuracy comparable to the one obtained with the finite element method. The superior efficiency of the new finite-difference scheme (FDM) is demonstrated for a few ρ-dependent one-dimensional potentials which are usually employed to model the electron states in free-standing and core–shell quantum wires. The new scheme is compared with the other FDM schemes for solving the effective-mass Schrödinger equation, and is found to deliver energy levels with much smaller numerical error for all the analyzed potentials. It also gives more accurate results than the scheme of Rizea et al., except for the ground state of an infinite rectangular potential in freestanding quantum wires. Moreover, the PT symmetry is invoked to explain similarities and differences between the considered FDM schemes.

Efficient high-order finite difference methods for nonlinear Klein–Gordon equations. I: Variants of the phi-four model and the form-I of the nonlinear Klein–Gordon equation

In this paper, the problem of solving some nonlinear Klein-Gordon equations (KGEs) is considered. Here, we derive different fourth- and sixth-order explicit and implicit algorithms to solve the phi-four equation and the form-I of the nonlinear Klein-Gordon equation. Stability and consistency of the proposed schemes are studied under certain conditions. Numerical results are presented and then compared with others obtained from some methods already existing in the scientific literature to explain the efficiency of the new algorithms. It is also shown that similar schemes can be proposed to solve many classes of nonlinear KGEs.

Steady-state ab initio laser theory for complex gain media.

We derive and test a generalization of the steady-state ab initio laser theory (SALT) to treat complex gain media. The generalized theory (C-SALT) is able to treat atomic and molecular gain media with diffusion and multiple lasing transitions, and semiconductor gain media in the free carrier approximation including fully the effect of Pauli blocking. The key assumption of the theory is stationarity of the level populations, which leads to coupled self-consistent equations for the populations and the lasing modes that fully include the effects of openness and non-linear spatial hole-burning. These equations can be solved efficiently for the steady-state lasing properties by a similar iteration procedure as in SALT, where a static gain medium with a single transition is assumed. The theory is tested by comparison to much less efficient finite difference time domain (FDTD) methods and excellent agreement is found. Using C-SALT to analyze the effects of varying gain diffusion constant we demonstrate a cross-over between the regime of strong spatial hole burning with multimode lasing to a regime of negligible spatial hole burning, leading to gain-clamping, and single mode lasing. The effect of spatially inhomogeneous pumping combined with diffusion is also studied and a relevant length scale for spatial inhomogeneity to persist under these conditions is determined. For the semiconductor gain model, we demonstrate the frequency shift due to Pauli blocking as the pumping strength changes.

Stress analysis of symmetric and anti-symmetric discretely stiffened laminated cantilever beams using displacement-potential field

The stress field of a laminated cantilever beam is analyzed under the influence of symmetric and anti-symmetric arrangements of discrete stiffeners at the opposing longitudinal surfaces. An efficient finite-difference computational scheme is developed, in which a new displacement potential is introduced to model the problem of laminated composites. Solutions of stresses at different plies of the laminated cantilever are obtained, some of which, especially those around the critical regions of stiffeners are presented. The effectiveness and accuracy of present approach is verified by comparing the results with the corresponding solutions of analytical as well as standard computational methods.

Mixed convection in gravity-driven nano-liquid film containing both nanoparticles and gyrotactic microorganisms

Analysis of a gravity-induced film flow of a fluid containing both nanoparticles and gyrotactic microorganisms along a convectively heated vertical surface is presented. The Buongiorno model is applied. Two kinds of boundary conditions, the passive and the active boundary conditions, are considered to investigate this film flow phenomenon. Through a set of similarity variables, the ordinary differential equations that describe the conservation of the momentum, the thermal energy, the nanoparticles, and the microorganisms are derived and then solved numerically by an efficient finite difference technique. The effects of various physical parameters on the profiles of momentum, thermal energy, nanoparticles, microorganisms, local skin friction, local Nusselt number, local wall mass flux, and local wall motile microorganisms flux are investigated. It is expected that the passively controlled nanofluid model can be much more easily achieved and applied in real circumstances than the actively controlled model.

An Efficient Compact Finite Difference Method for the Solution of the Gross-Pitaevskii Equation

We present an efficient, unconditionally stable, and accurate numerical method for the solution of the Gross-Pitaevskii equation. We begin with an introduction on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem. Then we present a new numerical method, CFDM-AIF method, which combines compact finite difference method (CFDM) in space and array-representation integration factor (AIF) method in time. The key features of our methods are as follows: (i) the fourth-order accuracy in space andrth (r≥2) accuracy in time which can be achieved and (ii) the significant reduction of storage and CPU cost because of array-representation technique for efficient handling of exponential matrices. The CFDM-AIF method is implemented to investigate the ground and first excited state solutions of the Gross-Pitaevskii equation in two-dimensional (2D) and three-dimensional (3D) Bose-Einstein condensates (BECs). Numerical results are presented to demonstrate the validity, accuracy, and efficiency of the CFDM-AIF method.

A finite-difference scheme for solution of a fractional heat diffusion-wave equation without initial conditions

Efficient finite-difference scheme to solve fractional diffusion-wave equations without initial conditions has been developed. The efficient approximation of the Riesz fractional derivatives is demonstrated and efficiently exemplified by two simple problems with/without source terms.

Numerical methods for high-power Er/Yb-codoped fiber amplifiers

We present effective methods for the numerical simulation of high-power-pumped continuous-wave (CW) and pulsed erbium–ytterbium-codoped fiber amplifiers (EYDFAs). For CW EYDFAs, a novel initial guess method and a robust algorithm are proposed to ensure the convergence of the boundary-value problems involved in the solution of the numerical model. For pulsed EYDFAs, an efficient finite-difference method based on the Lax–Wendroff scheme is developed. With these methods, high-power CW or pulsed EYDFAs can be simulated. As an example, a high-power pulse-pumped low repetition rate EYDFA is analyzed. Besides demonstrating the proposed methods, the simulation results show that the optimal pulse width of the pump can be determined by the power of the backward Yb–ASE, and the time delay between the signal and pump pulses play an important role in optimizing the performance of a high power pulse-pumped EYDFA.

On a potential‐velocity formulation of incompressible Navier‐Stokes equations

AbstractComputational fluid dynamics has emerged as an essential investigative tool in nearly every field of technology. Despite a well‐developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging, especially due to the non‐linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably.Recently, by introduction of an auxiliary potential field, a first integral of the two‐dimensional incompressible Navier‐Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original equations [1]. A useful application to free surface simulation was found in [2]. Moreover the new formulation is naturally extendible to three dimensions via tensor calculus, involving a non‐unique symmetric tensor potential. The corresponding degrees of freedom can be used in order to achieve a numerically convenient representation. Finally an efficient staggered‐grid finite difference scheme is applied to a Stokes flow problem in a 3D lid‐driven cavity to demonstrate the capabilities of the new approach. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new parallel meshing technique integrated into the conformal FDTD method for solving complex electromagnetic problems

A new efficient parallel finite-difference time-domain (FDTD) meshing algorithm, based on the ray tracing technique, is proposed in this paper. This algorithm can be applied to construct various FDTD meshes, such as regular and conformal ones. The Microsoft F# language is used for the algorithm coding, where all variables are unchangeable with its parallelization advantage being fully exploited. An improved conformal FDTD algorithm, also integrated with an improved surface current algorithm, is presented to simulate some complex 3D models, such as a sphere ball made of eight different materials, a tank, a J-10 aircraft, and an aircraft carrier with 20 aircrafts. Both efficiency and capability of the developed parallel FDTD algorithm are validated. The algorithm is applied to characterize the induced surface current distribution on an aircraft or a warship.

Dynamic skin deformation using finite difference solutions for character animation

We present a new skin deformation method to create dynamic skin deformations in this paper. The core elements of our approach are a dynamic deformation model, an efficient data-driven finite difference solution, and a curve-based representation of 3D models. We first reconstruct skin deformation models at different poses from the taken photos of a male human arm movement to achieve real deformed skin shapes. Then, we extract curves from these reconstructed skin deformation models. A new dynamic deformation model is proposed to describe physics of dynamic curve deformations, and its finite difference solution is developed to determine shape changes of the extracted curves. In order to improve visual realism of skin deformations, we employ data-driven methods and introduce skin shapes at the initial and final poses into our proposed dynamic deformation model. Experimental examples and comparisons made in this paper indicate that our proposed dynamic skin deformation technique can create realistic deformed skin shapes efficiently with a small data size.

Numerical algorithms for research and development stochastic control models

We consider the optimal strategy of research and development (R&D) expenditure adopted by a firm that engages in R&D to develop an innovative product to be launched in the market. The firm faces technological uncertainty associated with the success of the R&D effort and market uncertainty about the stochastic revenue flow generated by the new product. Our model departs from most R&D models by assuming that the firm’s knowledge accumulation has an impact on the R&D process, so the hazard rate of arrival of R&D success is no longer memoryless. Also, we assume a finite life span of the technologies that the product depends on. In this paper, we propose efficient finite difference schemes that solve the Hamilton–Jacobi–Bellman formulation of the resulting finite time R&D stochastic control models with an optimal control on R&D expenditure and an optimal stopping rule on the abandonment of R&D effort. The optimal strategies of R&D expenditure with varying sets of model parameters are analyzed. In particular, we observe that R&D expenditure decreases with a firm’s knowledge stock and may even drop to zero when the accumulation level is sufficiently high.