Second-order Difference Scheme Research Articles

Difference Schemes on Nonuniform Grids for the Two-Dimensional Convection–Diffusion Equation

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.

Numerical investigation of turbulent swirling flows through an abrupt expansion tube

A numerical investigation of turbulent swirling flows through an abrupt expansion tube is reported. The TEFESS code, based on a staggered Finite Volume approach with the standard k-ε model and first-order numerical schemes built-in, was used to carry out all the computations. The code has been modified in the present work to incorporate the ASM and two second-order numerical schemes. The ASM, which includes the non-gradient convection terms arising from the transformation from Cartesian to cylindrical coordinates, was investigated for isothermal flows by applying it to the flow through an abrupt expansion tube with or without swirl flows. In addition, to investigate the effects of numerical diffusion on the predicted results, two second-order differencing schemes, namely, second-order upwind and the quadratic upstream interpolation, were used to compare with the first-order hybrid scheme. An abrupt expansion tube with non-swirling flow, predicted results using both the k-ε model and the ASM were in good agreement with measurements. For swirling flows, the calculated results suggested that the use of the ASM with a second-order numerical scheme leads to better agreement between the numerical results and experimental data, while the k-ε model is incapable of capturing the stabilizing effect of the swirl.

Stability and convergence of difference schemes for multi-dimensional parabolic equations with variable coefficients and mixed derivatives

ABSTRACTWe present second-order difference schemes for a class of parabolic problems with variable coefficients and mixed derivatives. The solvability, stability and convergence of the schemes are rigorously analysed by the discrete energy method. Using the Richardson extrapolation technique, the fourth-order accurate numerical approximations both in time and space are obtained. It is noted that the Richardson extrapolation algorithms can preserve stability of the original difference scheme. Finally, numerical examples are carried out to verify the theoretical results.

A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer

In this paper, we consider a singularly perturbed reaction-diffusion problem with a discontinuous source term. Boundary and interior layers appear in the solution. The problem is discretized by using a hybrid finite difference scheme on a Shishkin-type mesh. A nonequidistant generalization of the Numerov scheme is used on the Shishkin-type mesh except for the point of discontinuity, whereas a second-order difference scheme with an additional refined mesh is used for the point of discontinuity. Although the difference scheme does not satisfy the discrete maximum principle, the maximum norm stability of the scheme is established. The maximum error in the mesh points is shown to be uniformly bounded by ( N^{-1}ln N ) ^{4} with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

• A coupled DDF-LB model was used to simulate high-speed compressible viscous flows. • NND and second-order upwind difference schemes were compared in simulations. • The DDF-LB model could simulate a compressible boundary layer up to Ma = 6. • Local high pressure appeared with the shock wave due to LBM’s mesoscopic feature. • High potential of LBM was shown for the shock boundary layer interaction problem. In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.

Parameter-uniform hybrid difference scheme for solutions and derivatives in singularly perturbed initial value problems

A second-order initial value problem with a small parameter multiplying the first- and second-order derivatives is considered. The precise knowledge about the behavior of the exact solution is analyzed. Based on the behavior of the exact solution, a hybrid finite difference scheme on a Shishkin mesh is proposed, which is a combination of the second-order difference scheme on the fine mesh and the modified midpoint upwind scheme on the coarse mesh. By applying the truncation error estimate techniques and a difference analogue of Gronwall’s inequality we prove that the scheme is almost second-order convergent for numerical solutions and scaled numerical derivatives. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACTThis paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.

A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation

In the present work, a compact difference scheme with convergence order O(τ2 + h4) is proposed for the fourth-order fractional sub-diffusion equation, where h and τ are space and temporal step length, respectively. The method is based on applying the L2−1σ formula to approximate the time Caputo fractional derivative and employing compact operator to approximate the spatial fourth-order derivative. Using the special properties of L2−1σ formula and mathematical induction method, we obtain the unconditional stability and convergence for our scheme by discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical examples are given to verify the theoretical analysis and efficiency of the new developed scheme.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper, a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element (MFE) method in space combined with L1-approximation and implicit second-order backward difference scheme in time. The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived. Finally, some numerical tests are shown to verify our theoretical analysis.

求解Allen-Cahn方程的两种高效数值格式

基于算子分裂思想，本文提出了求解Allen-Cahn方程的两种高效算子分裂格式。首先将此方程分裂为线性项与非线性项两个部分：对线性部分，通过二阶中心差分法与四阶紧致差分法分别进行数值计算，并利用傅里叶方法给出了稳定性的分析；对非线性部分进行精确求解。最后，通过数值算例比较两种算子分裂格式的数值解差异以及运行效率，而且验证了两种格式满足能量递减规律。 Based on the idea of operator splitting, this paper proposes two efficient operator splitting schemes for the Allen-Cahn equation. The original equation is divided into linear and nonlinear parts. The linear part is approximated numerically by two different schemes: the second-order center difference scheme and the fourth-order compact difference scheme. The stability analysis of both schemes is discussed according to a Fourier stability analysis. The nonlinear part is solved accurately. Numerical comparisons are carried out to verify the accuracy and efficiency of the proposed methods, and to verify the given schemes satisfied the law of decreasing energy.

A transient one-dimensional numerical model for kinetic Stirling engine

A third-order numerical model based on one-dimensional computational fluid dynamics is developed for kinetic Stirling engines. Various loss mechanisms in Stirling engines, including gas spring hysteresis loss, shuttle loss, appendix displacer gap loss, gas leakage loss, finite speed loss, piston friction loss, pressure drop loss, heat conduction loss, mechanical loss and imperfect heat transfer, are considered and embedded into the basic control equations. The non-equilibrium thermal model is adopted for the regenerator to capture the oscillating features of the gas and solid temperatures. To improve the numerical stability and accuracy, the implicit second-order time difference scheme and the second-order upwind scheme are adopted for discretizing the time differential terms and convective terms, respectively. Experimental validations are then conducted on a beta-type Stirling engine with the extensive experimental data for diverse working conditions. The results show that the developed model has better accuracies than the previous second-order models. Good agreements are achieved for predicting various critical system parameters, including pressure-volume diagram, indicated power, brake power, indicated efficiency, brake efficiency and mechanical efficiency. In particular, both the experiments and simulations show that the Stirling engine charged with helium tends to have much lower optimal working frequencies and poorer performances compared to the hydrogen system. Based on the analyses of the losses, it reveals that the pressure drop in the flow channels plays a critical role in shaping the different behaviors. The pressure drop in the helium system is much larger and more sensitive to the frequency increase due to the much larger viscosity of gaseous helium. Hydrogen is a superior working gas for a Stirling engine. The transient characteristics of the oscillating flow and the associated thermal interactions between gas and solid in the regenerator are finally analyzed in order to have an insight of the complex thermodynamic process. The study provides a promising numerical approach in simulating Stirling engines for further understandings of their operating characteristics and the underling mechanisms.

A second-order difference scheme for the time fractional substantial diffusion equation

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully discrete Grünwald–Letnikov-formula-based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions. In addition, the problem with nonsmooth solution is also discussed, and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative. Numerical examples show the reliability and efficiency of the scheme.

Second-order approximation scheme combined with [formula omitted]-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

In this article, a second-order approximation scheme combined with an H1-Galerkin mixed finite element (MFE) method for solving nonlinear convection–diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H1-Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second-order weighted and shifted Grünwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L2 and H1-norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used L1-approximation with (2−α)th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

Точность численного решение уравнения диффузии-конвекции на основе разностных схем второго и четвертого порядков погрешности аппроксимации

The paper deals with the scheme of the second and fourth order approximation error for solving convection-diffusion problems. To model initial boundary value problem in the case when the functions of the right and the initial condition can be represented by finite sums of Fourier series in the trigonometric basis, we investigated the accuracy of difference schemes. It was found that the accuracy of the numerical solution depends on the number of units attributable to half the wavelength corresponding to the most high frequency harmonics in the final sum of the Fourier series, necessary to describe the behavior of calculated objects. The dependence of the diffusion approximation error terms difference schemes of second and fourth order of accuracy of the number of nodes. The comparison of the calculation results of two-dimensional convectiondiffusion problems and tasks of the Poisson-based schemes of the second and fourth order accuracy. In the expediency of transition to a scheme of high accuracy for solving applied problems of the estimates and is easy to obtain the numerical values of the gain in computation time by using schemes of higher order accuracy.

Numerical Study of the Effect of the Reynolds Numbers on Thermal and Hydrodynamic Parameters of Turbulent Flow Mixed Convection Heat Transfer in an Inclined Tube

A numerical investigation of the effect of the Reynolds number on the thermal and hydrodynamic parameters of mixed convection heat transfer of the water-Al2O3 nanofluid turbulent flow in an inclined circular channel is the subject of this article. The upper wall of the channel was under non-uniform heat flux and its lower part was isolated. The two-phase mixture model, the finite volume method, and the second-order upstream difference scheme were used to solve the governing equations. After reviewing the results, it was found that by increasing Reynolds number, the convective heat transfer coefficient and shear stress increase, but the surface friction coefficient decreases. In the case of Nusselt number and the surface friction coefficient, some equations were extracted.

On the possibility of using recursive trigonometric functions to calculate the distribution of nonequilibrium minority charge carriers in a two-layer semiconductor material

Some possibilities of using a new method for the continuous approximation of step functions are considered. It is based on using trigonometric expressions in the form of recursive functions. For a model of independent sources, a second-order difference scheme of approximation on a uniform mesh is constructed. This makes it possible to calculate the distribution of minority charge carriers generated by a broad electron beam in a two-layer semiconductor material.

Nonlinear Vibration of PZT4/PZT-5H Monomorph and Bimorph Beams with Graded Microstructures

Monomorph and bimorph actuators made of piezoelectric materials are widely used in smart materials and structural control. Recent studies show that their performances can be remarkably improved by the use of functionally graded piezoelectric materials (FGPMs) whose compositional profile and effective material properties are graded along the thickness direction. This paper investigates the linear and nonlinear vibration behaviors of monomorph and bimorph made from a mixture of PZT4 and PZT-5H with material composition following a power law distribution. Theoretical formulations are based on von Kármán kinematic relationships and Timoshenko beam theory to account for the effect of transverse shear deformation. The differential quadrature method (DQM) and the second-order backward difference scheme are employed to obtain the linear and nonlinear vibration frequencies. A parametric study is conducted to investigate the influences of volume fraction index, slenderness ratio, nonlinear deformation and different loading conditions.

Optimal difference schemes for Maxwell’s equations in solving forward problems of electromagnetic soundings

In this paper, the solution of two-dimensional Maxwell’s equations is considered using the Laguerre transform. Optimal parameters of the difference schemes for the equations are obtained and presented. Numerical values of these optimal parameters are given. Second-order difference schemes with the optimal parameters provide an accuracy of the solution of the equations that is comparable to the accuracy of the solution using fourth-order schemes. It is shown that, when using the Laguerre transform, the number of optimal parameters can be reduced compared to the Fourier transform. This reduction leads to a simplification of the difference scheme and a reduction in the amount of computation, i.e., to efficiency of the algorithm.

The Effect of Reynolds Number on the Thermal and Hydrodynamic Characteristics of Turbulence Flow of the Nanofluid in the Heat Exchanger

Abstract. Reynolds number is one of the most important parameters in investigation of heat transfer in double tube heat exchangers. In this paper, the effect of this parameter has been investigated on the convective heat transfer coefficient and surface friction coefficient of the wall. Turbulent forced convection heat transfer of nanofluid flow of Al 2 O 3 /water in a double tube heat exchanger with rough tubes in the annular portion was numerically studied. The finite volume method and the second-order upstream difference scheme were used for the discretization of the governing equations. The volume fraction and mean diameter of nanoparticles were assumed to be 4% and 32 nm, respectively. After reviewing the results, it was observed that the convective heat transfer coefficient in the outer and inner walls of heat exchanger increases with the increase of the Reynolds number. The wall surface friction coefficient, which decreases by increasing Reynolds number, is another examined parameter.