High Accuracy Numerical Method Research Articles

Influence of magnetic field on dynamic behavior of double-diffusive convection

Double-diffusive convection under an external magnetic field has several industrial applications, and how the magnetic field affects flow and heat and mass transfer is a research focus. This paper studies the dynamical behavior of two-dimensional double-diffusive convection in a uniform magnetic field is investigated using a high-accuracy numerical method. The effects of the magnetic field strength on the flow and on the heat and mass transfer are analyzed, and the process of flow transition in the presence of a magnetic field is studied. The results show that, as the strength of the magnetic field increases, the flow is gradually inhibited, and most of the heat transfer occurs through conduction. For a fixed magnetic field strength, the temperature-dominated flow changes from periodic to chaotic, and finally back to periodic, as the orientation of the magnetic field increases. Two types of periodic flow occur in the transition route, namely, a simple periodic flow with an integer multiple of the fundamental frequency and a complex periodic flow with multiple lower-frequency subharmonic waves. In the presence of a strong magnetic field, the effect of the magnetic field orientation on the flow is significant. At high Rayleigh numbers and a fixed magnetic field strength, the heat transfer efficiency of a vertical magnetic field surpasses that of a horizontal magnetic field by approximately 50% when temperature is dominant, achieving a mass transfer efficiency of about 100%. In cases when concentration is dominant, the heat transfer efficiency of a vertical magnetic field is 20% higher than that of a horizontal magnetic field, and the mass transfer efficiency is approximately 60% higher for the vertical magnetic field compared to the horizontal magnetic field. The flow transitions from chaotic or periodic to steady as the Prandtl number increases. The solution branch bifurcates at certain Prandtl numbers, and hysteresis appears when Prandtl number is equal to 0.023. For low Rayleigh numbers, the heat and mass transfer efficiencies increase monotonically with increasing Prandtl number under a weak magnetic field but remain constant for various Prandtl numbers under a strong magnetic field.

A High Accuracy Numerical Method Based on Interpolation Technique for Time-Fractional Advection-Diffusion Equations

In this paper, the time-fractional advection-diffusion equation (TFADE) is solved by the barycentric Lagrange interpolation collocation method (BLICM). In order to approximate the fractional derivative under the definition of Caputo, BLICM is used to approximate the unknown function. We obtain the discrete scheme of the equation by combining BLICM with the Gauss-Legendre quadrature rule. The convergence rate for the TFADE equation of the BLICM is derived, and the accuracy of the discrete scheme can be improved by modifying the number of Gaussian nodes. To illustrate the efficiency and accuracy of the present method, a few numerical examples are presented and compared with the other existing methods.

Analysis of a High-Accuracy Numerical Method for Time-Fractional Integro-Differential Equations

A high-order finite difference numerical scheme based on the compact difference operator is proposed in this paper for time-fractional partial integro-differential equations with a weakly singular kernel, where the time-fractional derivative term is defined in the Riemann-Liouville sense. Here, the stability and convergence of the constructed compact finite difference scheme are proved in L∞ norm, with the accuracy order O(τ2+h4), where τ and h are temporal and spatial step sizes, respectively. The advantage of this numerical scheme is that arbitrary parameters can be applied to achieve the desired accuracy. Some numerical examples are presented to support the theoretical analysis.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

Comparison between the RANS Simulations of Double-Body Flow and Water–Air Flow around a Ship in Static Drift and Circle Motions

Manoeuvrability is one of the important ship hydrodynamic performances. That is closely related to the safety and economy of navigation. The development of a high-accuracy and high-efficiency numerical method to compute the forces and moments on manoeuvring ships has been the main task for ship manoeuvring predictions. The numerical method by solving RANS (Reynolds-Averaged Navier–Stokes) equations may be the most used one nowadays for the computations of ship manoeuvring forces and moments. However, applying a RANS tool for ship manoeuvring prediction remains very low efficiency, especially considering the six-degrees-of-freedom ship motions on the water surface. Thus, it is very necessary to introduce a few assumptions to reduce the computational time when applying a RANS tool, e.g., the assumptions of double-body flow and body force propeller, and consequently improve the application efficiency. Generally speaking, the assumption of double-body flow, in which the free-surface effects are neglected, is more suitable for low-speed ships. Nevertheless, rare publications have been reported relating to how this assumption affects the accuracy of the computed manoeuvring forces and moments. To this end, this article presents a comparative study between the RANS simulations of double-body flow and water–air flow around a container ship performing static drift and static circle motions. Three ship speeds, corresponding to the Froude numbers 0.156, 0.201, and 0.260, respectively, are considered during the simulations. The computed side forces and yaw moments obtained by the water–air flow simulations are closer to the available experimental data than that obtained by the double-body flow simulations for all ship speeds. The computed surge forces obtained by water–air flow simulations also agree well with the experimental data, whereas the computed surge forces obtained by the double-body flow simulations are wrong. The reasons are analyzed by comparing the pressure distributions on the ship surface and the flow separations around the ship.

Numerical investigation of 2D double-diffusive convection in rectangular cavities with different aspect ratios: Heat and mass transfer and flow characteristics

In this paper, the dynamical behavior of two-dimensional double-diffusive convection is numerically investigated using a high-accuracy numerical method. The process of flow transition in the presence of buoyancy is studied in detail, and the effects of the fluid properties and geometric parameters on the flow characteristics and heat and mass transfer are discussed. The results show that, as the buoyancy ratio increases from 0 to 2, the flow undergoes a complex series of transitions, from a steady, temperature-dominated state to periodic motion, then chaotic motion, back to periodic motion, and finally back to a steady, concentration-dominated state. At a fixed buoyancy ratio, when the Prandtl number Pr is less than 1, the flow changes from periodic or chaotic to steady with increasing Pr, and the heat and mass transfer efficiencies oscillate with an increasing trend. When Pr≥1, the flow is steady, and the heat and mass transfer remain nearly constant. For low Rayleigh numbers, the heat and mass transfer efficiencies increase monotonically with increasing Lewis number, but the flow is always in a steady state. For high Rayleigh numbers, the flow transitions from steady to periodic or chaotic via a supercritical Hopf bifurcation with increasing Lewis number, and the heat and mass transfer efficiencies oscillate with an increasing trend. In the range of aspect ratios considered in this study, the heat and mass transfer efficiencies exhibit an overall decay with increasing aspect ratio.

A multi-temporal series high-accuracy numerical manifold method for transient thermoelastic fracture problems

This paper presents a multi-temporal series high-accuracy numerical manifold method (NMM) for fracture analysis of solid materials under highly time-dependent thermal loadings. In the present method, the framework of NMM is used to perform space discretization of the thermo-mechanical coupling system, due to its advantages in accurately calculating field variables and describing discontinuity. An explicit direct integration scheme based on the precise time step integration method (or PTSIM scheme for short) is constructed for discretization in the time domain. In the present PTSIM scheme, the time-dependent thermal loadings are expressed by means of the basic function in a polynomial form. Highly accurate calculation of the response matrix for the load item is achieved by introducing the exponential transfer matrix. The results of stability, error and convergence analyses indicate that the proposed method is unconditionally stable and convergent. The superior advantages of present method are firstly verified by one example for transient thermal analysis, then four numerical examples with different time-dependent thermal boundary conditions and crack configurations are used for fracture analysis. The results show that the present method is unconditionally stable and time-independent, and high-accuracy can be achieved even for highly time-dependent thermal loadings and large time-step sizes.

Numerical investigation of double-diffusive convection in rectangular cavities with different aspect ratio I: High-accuracy numerical method

In this paper, being based on the idea of dispersion-relation-preserving optimization, a class of high-order high-resolution upwind compact schemes with a free parameter and their consistent boundary scheme are proposed for the solution of the governing equations of the 2D double-diffusive convection. The first derivative and the second derivative, in the vorticity equation, the temperature equation and the concentration equation, are discretized by using the optimal upwind compact scheme on a uniform mesh and the fourth-order symmetrical Padé compact scheme, respectively. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point two dimensional stencil. The fourth-order Runge-Kutta method is utilized for the temporal discretization. To assess numerical capability of the proposed algorithm, particularly its spatial behavior, the problems about the convection diffusion equation, the nonlinear Burgers equation and the nature convection flows in the square cavity with adiabatic horizontal walls and differentially heated vertical walls are computed by using the newly proposed algorithm. The numerical results are in excellent agreement with the benchmark solutions and some of the accurate results available in the literature, and show that effectiveness, accuracy and the advantage of better resolution of the present method. After that, steady and unsteady solutions for the double-diffusive convection in a rectangular cavity are also used to assess the efficiency of the present algorithm. The period of oscillation and flow field profiles are in great agreement with the data in the literature for buoyancy ratio, λ=1. The typical separation and secondary vortices at the bottom corner of the cavity as well as the top corner can be captured well for various buoyancy ratio. These indicate that the present method is suitable for simulating effectively the unsteady double-diffusive convection.

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Vibrational frequencies of FG-GPLRC viscoelastic rectangular plate subjected to different temperature loadings based on higher-order shear deformation theory and utilizing GDQ procedure

In this study, the effect of nonlinear temperature gradient on the natural frequencies of functionally graded graphene platelets reinforced composite (FG-GPLRC) viscoelastic plate embedded in the visco-Pasternak foundation is examined on the basis of higher-order shear deformation theory (HSDT) numerically and analytically. By performing Hamilton’s principle, the governing equations of the system are obtained. And also, the viscoelastic properties of the structure are modeled based on the appropriate Kelvin–Voigt viscoelasticity model. Afterward, the deflection problem of the system as a function of time is solved according to the fourth-order Runge–Kutta numerical manner. Also, in order to obtain the numerical results, the generalized differential quadrature method (GDQM) is applied as a high accuracy numerical method. For validation, the calculated numerical results are compared with those reported in the engineering literature. Furthermore, to demonstrate the effects of various parameters on the response of the system, such as various patterns of temperature rise, the stiffness of the foundation, damper, and viscoelasticity constants, weight fraction, and distribution patterns of GPLs, a comprehensive parametric study is done. According to the results, an increment in the temperature difference leads to an increase in the absolute value of sensitivity for various distributions of temperature. Ascending the nonlinearity of the temperature distribution soars the rigidity of the system; consequently, contrary to the dimensionless vibrational frequency lessens the transverse displacement of the structure.

Effect of inclination on nonlinear evolution and bifurcation of thermal convection in a square cavity

Heat transfer of natural convection in inclined cavities is one of the hot research topics in nonlinear non-equilibrium systems. In this paper, direct numerical simulations of natural convection in an inclined square cavity are carried out by using a high-accuracy numerical method. The effects of the different trends of inclination angle in a range of 0°–180° on the nonlinear evolution of flow field, heat transfer efficiency, and bifurcation are investigated. The Rayleigh number varies in a range from 10<sup>3</sup> to 10<sup>6</sup>. The results show that the heat transfer efficiency characterized by Nusselt number is highly dependent on the Rayleigh number, Prandtl number, and the inclination angle. When the Rayleigh number is high, the Nusselt number will have a small jump near the inclination angle in a range of 80°–100°. The evolution of the flow field and temperature field are more complicated at high Rayleigh number. There are one to three vortices of different intensities in the cavity. At low Rayleigh number and inclination angle of the cavity being close to 90°, the flow state is composed mainly of heat conduction state. In addition, it is found that there exist two stable branches of solutions in a range of Rayleigh number (4949, 314721) when the inclination angle is in the interval of (70°, 110°).

The reweighted Landweber scheme for the extrapolation of band-limited signals-II

The extrapolation of a band-limited signal is in terms of its finite known segment in the time interval [−T,T] for T > 0 to determine the unknown part of the signal. Some of the singular values of the operator of the reconstruction equation are close to 1 while the rest are close to zero, and if T is small, more singular values are close to zero. In this paper, we theoretically improve the posedness of the extrapolation of a band-limited signal by using a reweighting method to improve the distribution properties of all eigenvalues in infinite dimensions and accordingly propose a reweighted Landweber scheme. In the implementation, a high accuracy numerical method of the operator of the equation for reconstruction is proposed to reduce the truncated errors of the reconstruction equation and thus the accumulative errors of the reweighting method. Numerical experiments validate the proposed method. Compared with our previous work and existing works, effective reconstructions are obtained with a much smaller T.

A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain

A High Accuracy Numerical Method and Error Analysis for Fourth Order Elliptic Eigenvalue Problems in Circular Domain

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

PurposeThis paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.Design/methodology/approachIn this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.FindingsThe paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.Research limitations/implicationsThere are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.Practical implicationsPhysical problems with singular and non-singular coefficients are directly solved by this method.Originality/valueThe paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options

This paper is concerned with the design of a high order numerical approach based on a uniform mesh for efficient numerical solution of time-fractional Black-Scholes equation, governing European options. The time-fractional derivative is defined in the Caputo sense. A collocation method based on quintic B-spline basis functions is used for space discretization and time-stepping is done using a backward Euler method. The stability and convergence of the method are analyzed. The method is shown to be unconditionally stable and fourth order accurate in space and (2−β) order accurate in time, where β is the order of the time fractional derivative (0<β<1). Two numerical examples with the known exact solutions are considered to validate theoretical results and demonstrate the accuracy of the method. Moreover, this scheme is used to price three different European options governed by a time-fractional Black-Scholes model; (i) European double barrier knock-out call option (ii) European call option and (iii) European put option. The effect of the order of fractional derivative on the option price is studied.

High-accuracy hull iteration method for uncertainty propagation in fluid-conveying carbon nanotube system under multi-physical fields

High-accuracy numerical method is of significance in nano-electromechanical systems. This paper focuses on the influence of nano-system uncertainties on the critical flow velocity of fluid-conveying carbon nanotubes (CNTs) under multi-physical fields. A hull iteration method is proposed to estimate the bounds of the critical flow velocity of fluid-conveying CNT. The nonlinear governing equations for a fluid-conveying CNT embedded in an elastic medium and subjected to thermal and magnetic fields are derived by using the nonlocal Euler–Bernoulli beam theory. Considering the limited available information on the uncertainties, the material and geometrical properties of the CNT are described by using unknown-but-bounded parameters. The iterative scheme is presented in the framework of interval mathematics. After the assessment of the proposed method by comparing Monte Carlo simulation and stochastic analysis method, the detailed parametric investigation is carried out to deeply understand the coupled effects of uncertainties and many parameters such as small scale, Knudsen number, elastic medium and magnetic field parameters, as well as temperature changes. Additionally, the effect of geometric nonlinearity on the critical flow velocity is also examined in detail. The results derived in this paper can provide useful insights on the further application of CNTs as nano-channels to convey fluids.

A novel iterative algorithm for natural frequency analysis of FG thin plates under interval uncertainty

High-accuracy numerical method for uncertainty propagation analysis is a crucial issue in the field of structural reliability design. In this work, a novel iterative algorithm is proposed to study the free vibration of the functionally graded (FG) thin plates with material uncertainties. As a highly accurate approach distinct from the existing non-iterative model, the frequency analysis of FG thin plates can be achieved by updating the lower and upper bounds of natural frequency step by step. Based on the classic plate theory, the governing equations of the FG thin plate resting on an elastic medium are derived and the analytical formulation for the natural frequency is presented. By introducing interval parameters to quantify the material uncertainties, a non-probabilistic model for evaluating the natural frequency response of the embedded FG thin plate is developed. Subsequently, the deduction of the novel iterative algorithm for solving the non-probabilistic model is given based on interval mathematics. The developed model and the proposed iterative algorithm are validated by the Monte Carlo method, and then the detailed parametric studies are carried out to explore the combined influences of material uncertainties, power-law index, and elastic foundation parameters, as well as size parameters on the natural frequency of the FG thin plate. Numerical results can provide useful guidance in the precise design of FG structures.

Calculation and analysis of corrected thermal resistance in a common dwelling building in Romania, in various thermal insulation options – The exterior wall

Abstract The present paper analyzes the value of the corrected thermal resistance of the exterior wall at a typical residential building in Romania. The study was conducted for 8 variants of thermal insulation, with a thickness between 5 and 40 cm. The calculus was performed with the CIMPSPAT software, which makes use of a high accuracy numerical method for the thermal balance written in the nodes of the calculus according to the provisions in SR EN ISO 10211:2008. When designing buildings from a thermal point of view, only the unidirectional thermal resistances are generally considered, without taking into account the regulations of the Norm C107/2010 and Order no. 2641/2017, which requires for different building elements (exterior walls, exterior carpentry, floors over the last level, ground slabs, attic floor), certain minimum values for the corrected thermal resistance, which also take into account the thermal bridges. Thus, for the exterior wall, we generally use a 10 cm thick extruded polystyrene insulation. In this case study on a typical residential building in Romania, we want to determine the minimum thickness of this thermal insulation that satisfies the requirements of the Norm C107/2010 and Order no. 2641/2017, which requires for the exterior wall to have a minimum value of the corrected thermal resistance of 1,80 m2·K/W

Research on Magnetic Field Measurement System Based on Distributed Magnetic Field Sensing and Numerical Integration Method

Magnetic field measurement is one of the key contents in the field of electromagnetic protection. In this paper, a magnetic field measurement system based on distributed magnetic sensing and a high-accuracy numerical integration method is proposed. And a magnetic field sensor with simple structure, small size, and wide measurement band is designed, which includes three magnetic field measurement coils that can realize the accurate measurement of a three-dimensional magnetic field. The output of the magnetic field measurement coil is a differential signal that needs to be reduced by the integrator. To solve the problems of traditional analog integrator and common digital integrator, this paper proposes a high-precision digital integral algorithm based on Gauss algorithm, which can effectively improve the accuracy of the integrator. Test results indicate that the error of the magnetic field measurement system is less than 5% in the range of 0.05–5T.

Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer

We are concerned with using high accuracy numerical methods for solving singularly perturbed problems whose solutions exhibit an interior layer. A rational spectral collocation method in barycentric form with sinh transformation is firstly applied to find the approximate solution to the given problem. The sinh transformation consists of two parameters: the location and the width of the interior layer. With the tool of the asymptotic analysis, the interior layer can be easily located. Then, in determining the width of the interior layer, we design a nonlinear unconstrained optimization problem which is solved by using the basic differential evolution (DE) algorithm. The feasibility and effectiveness of the proposed method is verified by numerical results. The proposed method can also be applied to other singularly perturbed problems. • We consider a high accuracy numerical method to solve singularly perturbed problems with an interior layer. • The presented method is based on the rational spectral collocation method in barycentric form and the DE algorithm. • We construct a nonlinear unconstrained optimization problem which is solved by using the DE algorithm. • It is noted that the method presented in this paper can be extended to other type of singularly perturbed problems.