Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

In this paper,two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation.Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense,the new schemes are proved to per-fectly preserve the total energy in the discrete sense.By using the standard energy method and the cut-off function technique,the optimal error estimates of the numerical solutions are established,and the convergence rates are of O(h^(4)+τ^(2))with mesh-size h and time-step τ.In order to improve the computational efficiency,an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step.The convergence of the iterative algorithm is also rigorously analyzed.Several numerical results are carried out to test the error estimates and conservative properties.

A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects,such as decreased accuracy and even numerical instability,of the entire computational method,especially for higher order methods.In this work,we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes(N-S)equations.In the pro-posed method,the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme.With such a feature,the nu-merical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme.Numerical examples show the effec-tiveness and accuracy of the present consistent compact high order scheme in L^(∞).Its application to two dimensional lid-driven cavity flow problem further exhibits that un-der the same condition,the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4^(th)order explicit scheme.The compact finite difference method equipped with the present consistent boundary technique im-proves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.

FOURTH-ORDER COMPACT SCHEMES FOR HELMHOLTZ EQUATIONS WITH PIECEWISE WAVE NUMBERS IN THE POLAR COORDINATES

In this paper, fourth-order compact finite difference schemes are proposed for solving Helmholtz equation with piecewise wave numbers in polar coordinates with axis-symmetric and in some cases that the solution depends both of independent variables. The idea of the immersed interface method is applied to deal with the discontinuities in the wave number and certain derivatives of the solution. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.

A HIGH-ORDER PAD SCHEME FOR KORTEWEG-DE VRIES EQUATIONS

A high-order finite difference Pade scheme also called compact scheme for solving Korteweg-de Vries （KdV） equations, which preserve energy and mass conservations, was developed in this paper. This structure-preserving algorithm has been widely applied in these years for its advantage of maintaining the inherited properties. For spatial discretization, the authors obtained an implicit compact scheme by which spatial derivative terms may be approximated through combining a few knots. By some numerical examples including propagation of single soliton and interaction of two solitons, the scheme is proved to be effective.

A FAST HIGH ORDER METHOD FOR ELECTROMAGNETIC SCATTERING BY LARGE OPEN CAVITIES

In this paper, the electromagnetic scattering from a rectangular large open cavity embedded in an infinite ground plane is studied. By introducing a nonlocal artificial boundary condition, the scattering problem from the open cavity is reduced to a bounded domain problem. A compact fourth order finite difference scheme is then proposed to discrete the cavity scattering model in the rectangular domain, and a special treatment is enforced to approximate the boundary condition, which makes truncation errors reach （.9（h4） in the whole computational domain. A fast algorithm, exploiting the discrete Fourier transfor- mation in the horizontal and a Gaussian elimination in the vertical direction, is employed, which reduces the discrete system to a much smaller interface system. An effective pre- conditioner is presented for the BICGstab iterative solver to solve this interface system. Numerical results demonstrate the remarkable accuracy and efficiency of the proposed method. In particular, it can be used to solve the cavity model for the large wave number up to 600π.

A Numerical Algorithm for Determination of a Control Parameter in Two-dimensional Parabolic Inverse Problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. In this study we present a high order scheme for determining unknown control parameter and unknown solution of two-dimensional parabolic inverse problem with overspe- cialization at a point in the spatial domain. In this approach, a compact fourth-order scheme is used to discretize spatial derivatives of equation and reduces the problem to a system of ordinary differential equations （ODEs）. Then we apply a fourth order boundary value method to the solution of resulting system of ODEs. So the proposed method has fourth order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes in the literature. Also we will investigate the effect of noise in data on the approximate solutions.

Numerical Approximation of Solution for the Coupled Nonlinear Schrodinger Equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O（τ2 ＋ h4） are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.

A Fast High Order Iterative Solver for the Electromagnetic Scattering by Open Cavities Filled with the Inhomogeneous Media

The scattering of the open cavity filled with the inhomogeneous media is studied.The problem is discretized with a fourth order finite difference scheme and the immersed interfacemethod,resulting in a linear system of equations with the high order accurate solutions in the whole computational domain.To solve the system of equations,we design an efficient iterative solver,which is based on the fast Fourier transformation,and provides an ideal preconditioner for Krylov subspace method.Numerical experiments demonstrate the capability of the proposed fast high order iterative solver.

A STUDY ON NUMERICAL METHOD OF NAVIER-STOKES EQUATION AND NON-LINEAR EVOLUTION OF THE COHERENT STRUCTURES IN A LAMINAR BOUNDARY LAYER

A new method for direct numerical simulation of incompressible Navier-Stokes equations is studied in the paper. The compact finite difference and the non-linear terms upwind compact finite difference schemes on non-uniform meshes in x and y directions are developed respectively. With the Fourier spectral expansion in the spanwise direction, three-dimensional N-S equation are converted to a system of two-dimensional explicit-implicit The treatment of equations. The third-order mixed scheme is employed the three-dimensional for time integration. non-reflecting outflow boundary conditions is presented, which is important for the numerical simulations of the problem of transition in boundary layers, jets, and mixing layer. The numerical results indicate that high accuracy, stabilization and efficiency are achieved by the proposed numerical method. In addition, a theory model for the coherent structure in a laminar boundary layer is also proposed, based on which the numerical method is implemented to the non-linear evolution of coherent structure. It is found that the numerical results of the distribution of Reynolds stress, the formation of high shear layer, and the event of ejection and sweeping, match well with the observed characteristics of the coherent structures in a turbulence boundary layer.