A Family of Fourth-Order and Sixth-Order Compact Difference Schemes for the Three-Dimensional Poisson Equation

In this paper a family of fourth-order and sixth-order compact difference schemes for the three dimensional (3D) linear Poisson equation are derived in detail. By using finite volume (FV) method for derivation, the highest-order compact schemes based on two different types of dual partitions are obtained. Moreover, a new fourth-order compact scheme is gained and numerical experiments show the new scheme is much better than other known fourth-order schemes. The outline for the nonlinear problems are also given. Numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference scheme.     

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

This 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.     

An EXCMG accelerated multiscale multigrid computation for 3D Poisson equation

Multiscale multigrid (MSMG) method is an effective computational framework for efficiently computing high accuracy solutions for elliptic partial differential equations. In the current MSMG method, compared to the CPU cost on computing sixth-order solutions by applying extrapolation and other techniques on two fourth-order solutions from different scales grids, much more CPU time is spent on computing fourth-order solutions themselves on coarse and fine grids, particularly for high-dimensional problems. Here we propose to embed extrapolation cascadic multigrid (EXCMG) method into the MSMG framework to accelerate the whole process. Numerical results on 3D Poisson equations show that the new EXCMG–MSMG method is more efficient than the existing MSMG method and the EXCMG method for sixth-order solution computation.     

An Extrapolation Cascadic Multigrid Method Combined with a Fourth-Order Compact Scheme for 3D Poisson Equation

Extrapolation cascadic multigrid (EXCMG) method is an efficient multigrid method which has mainly been used for solving the two-dimensional elliptic boundary value problems with linear finite element discretization in the existing literature. In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the \(L^2\)-norm and \(L^{\infty }\)-norm for the solution and its gradient when the exact solution belongs to \(C^6\). Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.     

Energy stable compact scheme for Cahn–Hilliard equation with periodic boundary condition

We present a compact scheme to solve the Cahn–Hilliard equation with a periodic boundary condition, which is fourth-order accurate in space. We introduce schemes for two and three dimensions, which are derived from the one-dimensional compact stencil. The energy stability is completely proven for the proposed scheme based on the application of the compact method and well-known convex splitting methods. Detailed proofs of the mass conservation and unique solvability are also established. Numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods.     

An optimal compact sixth-order finite difference scheme for the Helmholtz equation

In this paper, we present an optimal compact finite difference scheme for solving the 2D Helmholtz equation. A convergence analysis is given to show that the scheme is sixth-order in accuracy. Based on minimizing the numerical dispersion, a refined optimization rule for choosing the scheme’s weight parameters is proposed. Numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters.     

A new fourth-order difference scheme for solving an N-carrier system with Neumann boundary conditions

In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods.     

A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.     

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.     

Non-polynomial sextic spline solution of singularly perturbed boundary-value problems

In this paper, we develop a generalized scheme based on non-polynomial sextic spline for the numerical solution of second-order singularly perturbed two-point boundary-value problems. The proposed method is second, fourth- and sixth-order accurate. Convergence analysis of the fourth-order method is briefly discussed. We show that the approximate solution obtained by the proposed method is better than existing spline methods. Numerical examples are given to illustrate the efficiency of our methods.     

A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.     

An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.     

Conservative compact difference scheme for the Zakharov–Rubenchik equations

In this article, a compact difference scheme is investigated to solve the Zakharov–Rubenchik equations in one dimension. The new scheme is proved to conserve the total mass and energy in the discrete sense. Rigorous error estimates are established for the new method with the help of an induction argument in energy space which show that the new scheme has second-order accuracy in time and fourth-order accuracy in space. Extensive numerical results are provided to verify our theoretical analysis, and show the accuracy and efficiency of the new scheme.     

一类四阶微积分方程的四阶差分格式

针对由吊桥模型而建立的四阶微积分方程,提出了四阶差分格式进行求解．对线性项采用紧格式进行离散,积分项则采用复化辛普森求积公式处理,再结合Newton型迭代法对方程进行求解．给出了差分格式解的存在性和收敛性的证明．数值结果表明格式的精度为O（h4）．     

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

In this paper, a combined compact finite difference method (CCD) together with alternating direction implicit (ADI) scheme is developed for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients. The proposed CCD-ADI method is second-order accurate in time variable and sixth-order accurate in space variable. For the linear hyperbolic equation, the CCD-ADI method is shown to be unconditionally stable by using the Von Neumann stability analysis. Numerical results for both linear and nonlinear hyperbolic equations are presented to illustrate the high accuracy of the proposed method.     

Recent Advances in the Study of a Fourth-Order Compact Scheme for the One-Dimensional Biharmonic Equation

It is well-known that non-periodic boundary conditions reduce considerably the overall accuracy of an approximating scheme. In previous papers the present authors have studied a fourth-order compact scheme for the one-dimensional biharmonic equation. It relies on Hermitian interpolation, using functional values and Hermitian derivatives on a three-point stencil. However, the fourth-order accuracy is reduced to a mere first-order near the boundary. In turn this leads to an ??almost third-order?? accuracy of the approximate solution. By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full (??optimal??) fourth-order convergence is attained. A number of numerical examples corroborate this effect.     

Numerical study of fourth-order linearized compact schemes for generalized NLS equations

The fourth-order compact approximation for the spatial second-derivative and several linearized approaches, including the time-lagging method of Zhang et al. (1995), the local-extrapolation technique of Chang et al. (1999) and the recent scheme of Dahlby et al. (2009), are considered in constructing fourth-order linearized compact difference (FLCD) schemes for generalized NLS equations. By applying a new time-lagging linearized approach, we propose a symmetric fourth-order linearized compact difference (SFLCD) scheme, which is shown to be more robust in long-time simulations of plane wave, breather, periodic traveling-wave and solitary wave solutions. Numerical experiments suggest that the SFLCD scheme is a little more accurate than some other FLCD schemes and the split-step compact difference scheme of Dehghan and Taleei (2010). Compared with the time-splitting pseudospectral method of Bao et al. (2003), our SFLCD method is more suitable for oscillating solutions or the problems with a rapidly varying potential.     

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.     

A CCD–ADI method for unsteady convection–diffusion equations

With a combined compact difference scheme for the spatial discretization and the Crank–Nicolson scheme for the temporal discretization, respectively, a high-order alternating direction implicit method (ADI) is proposed for solving unsteady two dimensional convection–diffusion equations. The method is sixth-order accurate in space and second-order accurate in time. The resulting matrix at each ADI computation step corresponds to a triple-tridiagonal system which can be effectively solved with a considerable saving in computing time. In practice, Richardson extrapolation is exploited to increase the temporal accuracy. The unconditional stability is proved by means of Fourier analysis for two dimensional convection–diffusion problems with periodic boundary conditions. Numerical experiments are conducted to demonstrate the efficiency of the proposed method. Moreover, the present method preserves the higher order accuracy for convection-dominated problems.     

A comparison of second- and sixth-order methods for large-eddy simulations

Large-eddy simulations of spatially developing planar turbulent jets are performed using a compact finite-difference scheme of sixth-order and an advective upstream splitting method-based method of second-order accuracy. The applicability of these solution schemes with different subgrid scale models and their performance for realistic turbulent flow problems are investigated. Solutions of the turbulent channel flow are used as an inflow condition for the turbulent jets. The results compare well with each other and with analytical and experimental data. For both solution schemes, however, the influence of the subgrid scale model on the time averaged turbulence statistics is small. This is known to be the case for upwind schemes with a dissipative truncation error, but here it is also observed for the high-order compact scheme. The reason is found to be the application of a compact high-frequency filter, which has to be used with strongly stretched computational grids to suppress high-frequency oscillations. The comparison of the results of the two schemes shows hardly any difference in the quality of the solutions. The second-order scheme, however, is computationally more efficient.