High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

High-order finite difference schemes for the solution of second-order BVPs

We introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

A compact difference scheme for fourth-order fractional sub-diffusion equations with Neumann boundary conditions

In this paper, a compact finite difference scheme with global convergence order $O(\tau^{2}+h^4)$ is derived for fourth-order fractional sub-diffusion equations subject to Neumann boundary conditions. The difficulty caused by the fourth-order derivative and Neumann boundary conditions is carefully handled. The stability and convergence of the proposed scheme are studied by the energy method. Theoretical results are supported by numerical experiments.     

New higher-order compact finite difference schemes for 1D heat conduction equations

In this paper, we present two higher-order compact finite difference schemes for solving one-dimensional (1D) heat conduction equations with Dirichlet and Neumann boundary conditions, respectively. In particular, we delicately adjust the location of the interior grid point that is next to the boundary so that the Dirichlet or Neumann boundary condition can be applied directly without discretization, and at the same time, the fifth or sixth-order compact finite difference approximations at the grid point can be obtained. On the other hand, an eighth-order compact finite difference approximation is employed for the spatial derivative at other interior grid points. Combined with the Crank–Nicholson finite difference method and Richardson extrapolation, the overall scheme can be unconditionally stable and provides much more accurate numerical solutions. Numerical errors and convergence rates of these two schemes are tested by two examples.     

Fourth-order compact solution of the nonlinear Klein-Gordon equation

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.     

Finite difference methods for a class of two-point boundary value problems with mixed boundary conditions

We discuss the construction of three-point finite difference aproximations for the class of two-point boundary value problems: [p(x)y′]′ = f(x, y), α₀y(a) - α₁y′(a) = A, β₀y(b) + β₁y′(b) = B.We first establish an identity from which general three-point finite difference approximations of various orders can be obtained. We then consider in detail obtaining fourth-order methods based on three evaluations of f. We obtain a family of fourth-order discretizations for the differential equations; appropriate discretizations for the boundary conditions are also obtained for use with fourth-order methods. We select the free parameters available in this discretizations which lead to a “simplest” fourth-order method. This method is described and its convergence is established; numerical examples are given to illustrate this new fourth-order method.     

A family of fourth-order parallel splitting methods for parabolic partial differential equations

A family of numerical methods which are L-stable, fourth-order accurate in space and time, and do not require the use of complex arithmetic is developed for solving second-order linear parabolic partial differential equations. In these methods, second-order spatial dderivatives are approximated by fourth-order finite-difference approximations, and the matrix exponential function is approximated by a rational approximation consisting of three parameters. Parallel algorithms are developed and tested on the one-dimensional head equation, with constant coefficients, subject to homogeneous and time-dependent boundary conditions. These methods are also extended to two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13 : 357–373, 1997     

A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates

In this paper, we extend our previous work (M.-C. Lai, A simple compact fourth-order Poisson solver on polar geometry, J. Comput. Phys. 182 (2002) 337–345) to 3D cases. More precisely, we present a spectral/finite difference scheme for Poisson equation in cylindrical coordinates. The scheme relies on the truncated Fourier series expansion, where the partial differential equations of Fourier coefficients are solved by a formally fourth-order accurate compact difference discretization. Here the formal fourth-order accuracy means that the scheme is exactly fourth-order accurate while the poles are excluded and is third-order accurate otherwise. Despite the degradation of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condition is needed. The resulting linear system is then solved by the Bi-CGSTAB method with the preconditioner arising from the second-order discretization which shows the scalability with the problem size.     

求解对流扩散反应方程的四阶混合紧致差分方法

针对一维对流扩散反应方程,基于对流扩散方程的四阶指数型紧致差分格式,以及一阶导数的四阶Padé公式,发展了一种高效求解对流扩散反应方程的混合型四阶紧致差分格式.数值实验结果验证了格式对于边界层问题或大雷诺数或大Pelect数的大梯度问题的求解的高精度和鲁棒性的优点.     

The combination of collocation,finite difference,and multigrid methods for solution of the two‐dimensional wave equation

In this article, we apply compact finite difference approximations of orders two and four for discretizing spatial derivatives of wave equation and collocation method for the time component. The resulting method is unconditionally stable and solves the wave equation with high accuracy. The solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. We employ the multigrid method for solving the resulted linear system. Multigrid method is an iterative method which has grid independently convergence and solves the linear system of equations in small amount of computer time. Numerical results show that the compact finite difference approximation of fourth order, collocation and multigrid methods produce a very efficient method for solving the wave equation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Fast and Robust Sixth Order Multigrid Computation for 3D Convection Diffusion Equation

We present a sixth-order explicit compact finite difference scheme to solve the three-dimensional (3D) convection-diffusion equation. We first use a multiscale multigrid method to solve the linear systems arising from a 19-point fourth-order discretization scheme to compute the fourth-order solutions on both a coarse grid and a fine grid. Then an operator-based interpolation scheme combined with an extrapolation technique is used to approximate the sixth-order accurate solution on the fine grid. Since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid-independent convergence rate for solving convection-diffusion equations with a high Reynolds number, we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency. Supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth-order compact (SOC) scheme, compared with the previously published fourth-order compact (FOC) scheme.     

Numerical algorithm for solving the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this paper, based on the second-order compact approximation of first-order derivative, the numerical algorithm with second-order temporal accuracy and fourth-order spatial accuracy is developed to solve the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative; the solvability, convergence, and stability of the numerical algorithm are analyzed in detail by algebraic theory and Fourier analysis, respectively; the numerical experiment support our theoretical analysis results.     

A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel

In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L₁ discrete formula and the Riemann–Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis.     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Compact mixed methods for convection/diffusion type problems

This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we search a fourth-order compact approximation of the primary solution field and flux. Based on Taylor series expansion, the proposed compact mixed formulation generalizes the work of Carey and Spotz [G.F. Carey, W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Meth. Eng. 13 (1997)]. We show that their fourth-order formulation corresponds to a particular case of our presented scheme, and we extend their work to variable diffusion and convection coefficients. Some numerical experiments are performed to demonstrate the fourth-order effective convergence rate.     

A fourth-order Magnus scheme for Helmholtz equation

For wave propagation in a slowly varying waveguide, it is necessary to solve the Helmholtz equation in a domain that is much larger than the typical wavelength. Standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications. A popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment. For a slowly varying waveguide, it is possible that the length of such a segment is much larger than the typical wavelength. To reduce memory requirements, it is advantageous to reformulate the boundary value problem of the Helmholtz equation as an initial value problem using a pair of operators. Such an operator-marching scheme can also be solved with the piecewise uniform approximation of the waveguide. This is related to the second-order midpoint exponential method for a system of linear ODEs. In this paper, we develop a fourth-order operator-marching scheme for the Helmholtz equation using a fourth-order Magnus method.     

A fourth-order finite difference scheme for two-dimensional nonlinear elliptic partial differential equations

A finite difference scheme for the two-dimensional, second-order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13-point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h⁴. Well-known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth-order convergence of the scheme. © 1995 John Wiley & Sons, Inc.