COMPACT FINITE DIFFERENCE-FOURIER SPECTRAL METHOD FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

High accuracy compact finite difference methods and their applications

Numerical simulation of complex flow fields with multi-scale structures is one of the most important and challenging branches of computational fluid dynamics. From linear analysis and numerical experiments it has been discovered that the higher-order accurate method can give reliable and efficient computational results, as well as better resolution of the complex flow fields with multi-scale structures. Compact finite difference schemes, which feature higher-order accuracy and spectral-like resolution with smaller stencils and easier application of boundary conditions, has attracted more and more interest and attention.

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.

UNIFORM ERROR BOUNDS OF A CONSERVATIVE COMPACT FINITE DIFFERENCE METHOD FOR THE QUANTUM ZAKHAROV SYSTEM IN THE SUBSONIC LIMIT REGIME

In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.

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.

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrodinger-Poisson System

We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.

Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

A COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEME FOR THE IMPROVED BOUSSINESQ EQUATION WITH DAMPING TERMS

In this paper, a compact finite difference method is presented for solving the initial boundary value problems for the improved Boussinesq equation with damping terms. The fourth-order equation can be transformed into a first-order ordinary differential system, and then, the classical Pad4 approximation is used to discretize spatial derivative in the non- linear partial differential equations. The resulting coefficient matrix for the semi-discrete scheme is tri-diagonal and can be solved efficiently. In order to maintain the same order of convergence, the classical fourth-order Runge-Kutta method is the preferred method for explicit time integration. Soliton-type solutions are used to evaluate the accuracy of the method, and various numerical experiments are designed to test the different effects of the damping terms.

Explicit High-Order Method to Solve Coupled Nonlinear Schrödinger Equations

Models of the coupled nonlinear Schrödinger equations submit various critical physical phenomena with a typical equation for optical fibres with linear refraction. In this article, we will presuppose the Compact Finite Difference method with Runge-Kutta of order 4 (explicit) method, which is sixth-order and fourth-order in space and time respectively, to solve coupled nonlinear Schrödinger equations. Many methods used to solve coupled nonlinear Schrödinger equations are second order in time and need to use extra-technique to rise up to fourth-order as Richardson Extrapolation technique. The scheme obtained is immediately fourth-order in one step. This approach is a conditionally stable method. The conserved quantities and the exact single soliton solution indicate the competence and accuracy of the article’s suggestion schemes. Furthermore, the article discusses the two solitons interaction dynamics.

A Compact Eulerian Interface-Capturing Algorithm for Compressible Multimaterial Elastic-Plastic Flows with Mie-Gr uneisen Equation of State

This paper presents an Eulerian diffuse-interface method using a high-order compact difference scheme for simulating elastic-plastic flows with the Mie–Gruneisen(MG)equation of state(EoS).For simulations of multimaterial problems,numerical errors were generated in the material discontinuities owing to inconsistent treatment of the convective terms.Based on the normal-stress-based mechanical equilibrium assumption for elastic-plastic solids,we introduce an improved form of the consistent localized artificial diffusivity(LAD)method to ensure an oscillation-free interface for velocity and normal stress.The proposed algorithm uses a hyperelastic model.A mixture type of the model system was formed by combining the conservation equations for the basic conserved variables,an equation of a unified deviatoric tensor describing solid deformation,and an additional set of equations for solving the material quantities in the MG EoS.Several one-and two-dimensional problems with various discontinuities,including the elastic-plastic Richtmyer–Meshkov instability,were considered for testing the proposed method.

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.

Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media using Compact High Order Schemes

In many problems,one wishes to solve the Helmholtz equation with variable coefficients within the Laplacian-like term and use a high order accurate method(e.g.,fourth order accurate)to alleviate the points-per-wavelength constraint by reducing the dispersion errors.The variation of coefficients in the equation may be due to an inhomogeneous medium and/or non-Cartesian coordinates.This renders existing fourth order finite difference methods inapplicable.We develop a new compact scheme that is provably fourth order accurate even for these problems.We present numerical results that corroborate the fourth order convergence rate for several model problems.

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.

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.

INTERACTIVE STUDY BETWEEN IDENTICAL COHERENT STRUCTURES IN THE WALL REGION OF A TURBULENT BOUNDARY LAYER

A theoretical model for identica l coherent structures in the wall region of a turbulent boundary layer was propo sed, using the idea of general resonant triad of the hydrodynamic stability. The evolution of the structures in the wall region of a turbulent boundary layer wa s studied by combining the compact finite differences of high numerical accuracy and the Fourier spectral hybrid method for solving the three dimensional Navier -Stokes equations. In this method, the third order mixed explicit-implicit sch eme was applied for the time integration. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, an d the sixth-order center compact schemes for the derivatives in spectral space were introduced, respectively. The fourth-order compact schemes satisfied by th e velocities and pressure in spectral space was derived. As an application, the method was implemented to the wall region of a turbulent boundary to study the e volution of identical coherent structures. It is found that the numerical result s are satisfactory.

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 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.