Compact Difference Approximation Research Articles

A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.

Compact difference schemes for multidimensional Klein–Gordon equations

Abstract. In this article, we consider a compact difference approximation of the schemes of order O(| h|4 + τ2), h = (h1, h2, ..., hp) for the Klein–Gordon equations in the multidimensional case. In studying the stability of these difference schemes, the theory of operator-difference schemes by A. A. Samarskii is used, and the strong stability of difference schemes is proved with respect to a small perturbation of the initial conditions, the right-hand side and the coefficients of the equations. The theoretical results are confirmed by test numerical calculations.

A Crank-Nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t=0. A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

Compact difference schemes for Klein–Gordon equation with variable coefficients

In this paper, we consider the compact difference approximation of the fourth and second-order schemes on a three-point stencil for Klein–Gordon equations with variable coefficients. Despite the linearity of the differential and difference problems, it is not possible in this case to apply the well-known results on the theory of stability of three-layer operator-difference schemes by A. A. Samarskii. The main purpose is to prove the stability with respect to the initial data and the right-hand side of compact difference schemes in the grid norms L 2 (W h ), W 1 2 (W h ), C (W h ). Using the method of energy inequalities, the corresponding a priori estimates, expressing the stability and convergence of the solution to the difference problem with the assumption h ≤ = h 0, h 0 = const, τ≥h is obtained. The conducted numerical experiment shows how Runge rule is used to determine the different orders of the convergence rate of the difference scheme in the case of two independent variables.

Compact difference schemes for Klein-Gordon equation

In this paper, we consider compact difference approximation of the fourth-order schemes for linear, semi-linear, and quasilinear Klein-Gordon equations. with respect to a small perturbation of initial conditions, right-hand side, and coefficients of the linear equations the strong stability of difference schemes is proved. The conducted numerical experiment shows how Runge rule is used to determine the orders of convergence of the difference scheme in the case of two independent variables.

Study of a Flame Kernel Evolution in a Turbulent Mixing Layer Using LES with a Laminar Chemistry Model

Temporal evolution of an ignition kernel in a spark ignited turbulent hydrogen–air mixing layer is studied using the large eddy simulation approach with an implicit treatment of the reaction source terms. The applied numerical code is based on a high-order compact difference approximation combined with a weighted essentially non-oscillatory scheme, which provide an accurate resolution of the small scale phenomena. The spark is modelled by an energy deposition model coupled with the enthalpy equation. Since the fuel and oxidiser streams move in the opposite directions the flow is dominated by shear stresses and vortical structures. The ignition follows three different scenarios depending on the spark location. An interaction between the developing flame kernel and large turbulent structure is analysed starting from the energy transfer up to a fully reacting state. The size and shape of the flames are correlated with the initial ignition scenario. A 3D visualisation of the transient position of the flame kernel shows its different spatio-temporal behaviour. It is observed that the flame can stay close to the initial position and spread equally in all directions or it can move far from the initial location and follow the evolving flow field. In the latter scenario two separate sub-stages are convincingly identified. Concerning the ignition probability (P_{ign}), when the spark is located in the immediate vicinity of the mixing layer, the P_{ign} on the fuel-rich side is higher. On the other hand, when the ignition occurs far from the mixing layer, the P_{ign} is significantly larger on the lean side. Unlike the local composition the impact of the strain rate of the velocity field was found to have very limited impact on the P_{ign}.

Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.

Parametric study of multi-armed jets

The paper presents the results of numerical simulations of excited incompressible jets with the transport of a passive scalar field. A forcing is obtained through an axial and helical velocity disturbance defined as sinusoidal waves changing in time and travelling in azimuthal direction along the jet border. It includes three control parameters, which determine its strength and temporal behaviour, an amplitude and two characteristic frequencies. The research is performed with the use of direct numerical simulations (DNS) and high-order numerical code based on the compact difference approximation and projection method for the pressure-velocity coupling. The obtained results show that when the axial and helical forcing terms are applied separately, their impact on the flow and passive scalar field is very limited and only quantitative. The situation changes drastically when both forcing terms are applied together. It is found that by a proper choice of the ratio of their frequencies, various type of multi-armed jets can be created, e.g., with 5, 8, or 13 branches. It is observed that the angle at which the branches disjoin from the main jet and an axial location where this phenomenon starts are related to the axial forcing frequency and also to the Reynolds number, to some extent. It is shown that the mixing efficiency analyzed through an evolution of the scalar field or entrainment can be controlled in a smooth way and in a relatively wide range.

A Crank-Nicolson-type compact difference method and its extrapolation for time fractional Cattaneo convection-diffusion equations with smooth solutions

A high-order Crank-Nicolson-type compact difference method is proposed for a class of time fractional Cattaneo convection-diffusion equations with smooth solutions. The convection coefficient of the equation may be spatially variable. A suitable transformation is adopted to transform the original equation into a reaction-diffusion equation, which is then discretized by a fourth-order compact difference approximation for the spatial derivative and by a second-order Crank-Nicolson-type difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability of the method and its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. A Richardson extrapolation algorithm, including its rigorous convergence analysis, is presented. This extrapolation algorithm improves the temporal accuracy of the computed solution to the third order. An application of the proposed method to the non-smooth solution which has a weak singularity at the initial time is also discussed by introducing a correction term. Numerical results demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.

High-order compact difference algorithm on half-staggered meshes for low Mach number flows

The paper presents a solution algorithm for variable density non-isothermal flows based on a high-order compact difference approximation combined with the projection method formulated on half-staggered meshes. In contrast to full-staggered grid arrangement, with the pressure and all scalar variables located in the cell centers, we shift only the pressure nodes while we keep the scalars and velocity components in the same nodes. This greatly simplifies the solution procedure as the interpolation between the nodes takes place only within the projection method. Derivative approximations and interpolation formulas are discretized using the high-order compact difference schemes up to 10th order in the central nodes and third/fourth order schemes near domain boundaries. The proposed algorithm is first verified in natural convection problems in square and rectangular wall bounded cavities for Rayleigh numbers Ra=3.4×105 and Ra=1.0×106 with maximal density ratio equals nine. The results are compared with published data for both steady and unsteady flow regimes. The performed simulations reveal that achievement of accurate results is conditioned by a mesh resolution in thermal boundary layers near the walls, and a way in which the governing equations are formulated, i.e., the conservative or non-conservative form. This turns out more important than a formal order of discretization method. Robustness of the proposed algorithm for the computations of turbulent flow is demonstrated based on the flow in a periodic channel at Reynolds number Reτ=200 with density ratio equal to two. In this case the profiles of mean velocity, temperature and their fluctuations are analyzed and they agree quite well with literature data. In all presented test cases the obtained pressure fields are smooth and without any oscillations observed when the collocated meshes are employed.

The study of a fourth-order multistep ADI method applied to nonlinear delay reaction–diffusion equations

In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L2- and H1-norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms.

Higher order finite difference method for the reaction and anomalous-diffusion equation

In this paper, our aim is to study the high order finite difference method for the reaction and anomalous-diffusion equation. According to the equivalence of the Riemann–Liouville and Grünwald–Letnikov derivatives under the suitable smooth condition, a second-order difference approximation for the Riemann–Liouville fractional derivative is derived. A fourth-order compact difference approximation for second-order derivative in spatial is used. We analyze the solvability, conditional stability and convergence of the proposed scheme by using the Fourier method. Then we obtain that the convergence order is O(τ2+h4), where τ is the temporal step length and h is the spatial step length. Finally, numerical experiments are presented to show that the numerical results are in good agreement with the theoretical analysis.

Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation

In this paper, four alternating direction implicit (ADI) schemes are presented for solving two-dimensional cubic nonlinear Schrödinger equations. Firstly, we give a Crank–Nicolson ADI scheme and a linearized ADI scheme both with accuracy O(Δt2+h2), with the same method, use fourth-order Padé compact difference approximation for the spatial discretization; two HOC-ADI schemes with accuracy O(Δt2+h4) are given. The two linearized ADI schemes apply extrapolation technique to the real coefficient of the nonlinear term to avoid iterating to solve. Unconditionally stable character is verified by linear Fourier analysis. The solution procedure consists of a number of tridiagonal matrix equations which make the computation cost effective. Numerical experiments are conducted to demonstrate the efficiency and accuracy, and linearized ADI schemes show less computational cost. All schemes given in this paper also can be used for two-dimensional linear Schrödinger equations.

A higher order compact finite difference algorithm for solving the incompressible Navier–Stokes equations

AbstractOn the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier–Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third‐order accurate in space. A third‐order accurate upwind compact difference approximation is used to discretize the non‐linear convective terms, a fourth‐order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth‐order compact difference approximation on a cell‐centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth‐order compact difference scheme constructed currently on the nine‐point 2D stencil. New fourth‐order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine‐point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid‐driven cavity flow are also used to assess the efficiency of this algorithm. Copyright © 2011 John Wiley & Sons, Ltd.

High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.

Compact difference approximation with consistent boundary condition*

For simulating multi-scale complex flow fields it should be noted that all the physical quantities we are interested in must be simulated well. With limitation of the computer resources it is preferred to use high order accurate difference schemes. Because of their high accuracy and small stencil of grid points computational fluid dynamics (CFD) workers pay more attention to compact schemes recently. For simulating the complex flow fields the treatment of boundary conditions at the far field boundary points and near far field boundary points is very important. According to authors' experience and published results some aspects of boundary condition treatment for far field boundary are presented, and the emphasis is on treatment of boundary conditions for the upwind compact schemes. The consistent treatment of boundary conditions at the near boundary points is also discussed. At the end of the paper are given some numerical examples. The computed results with presented method are satisfactory.

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.