Fourth-order Finite Difference Scheme Research Articles

Numerical simulation of flow-induced cavity noise in self-sustained oscillations

The generation and near-field radiation of aerodynamic sound from a low-speed unsteady flow over a two-dimensional automobile door cavity is simulated by using a source-extraction-based coupling method. In the coupling procedure, the unsteady cavity flow field is first computed solving the Reynolds- averaged Navier–Stokes (RANS) equations. The radiated sound is then calculated by using a set of acoustic perturbation equations with acoustic source terms which are extracted from the time-dependent solutions of the unsteady flow. The aerodynamic and its resulting acoustic field are computed for the Reynolds number of 53,266 based on the base length of the cavity. The free stream flow velocity is taken to be 50.9 m/s. As first stage of the numerical investigation of flow-induced cavity noise, laminar flow is assumed. The CFD solver is based on a cell-centered finite volume method. A dispersion-relation-preserving (DRP), optimized, fourth-order finite difference scheme with fully staggered-grid implementation is used in the acoustic solver.

Fitted fourth-order tridiagonal finite difference method for singular perturbation problems

In this paper, a fitted fourth-order tridiagonal finite difference scheme is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. We have taken a fourth-order tridiagonal finite difference scheme by M.M. Chawla [A fourth-order tridiagonal finite difference method for general nonlinear two-point boundary value problems with mixed boundary conditions, J. Inst. Maths Appl. 21 (1978) 83–93] and introduced a fitting factor. The fitting factor is obtained from the theory of singular perturbations. Thomas Algorithm is used to solve the system. To demonstrate the applicability of the present method, we have solved five linear problems (three with left end and two with right end boundary layers). Solutions of these problems using the present fitted method are compared with Chawla’s solutions. From the results, it is observed that the present method is stable and has better approximation to the exact solution.

Numerical simulation of flow behind active vortex generators with direct forcing immersed boundary method

The immersed boundary method with discrete direct forcing approach was combined with a direct numerical simulation (DNS) code to study the time-dependent response of boundary-layer flow over a flat plate to the active vortex generators (AVG) that consist of a pair of deployable circular wing lip type blades driven by a pre-defined duty cycle. The DNS code solves the three-dimensional Navier–Stokes equations for compressible flow in general curvilinear coordinates using a fully implicit LU-SGS method. A fourth-order finite difference scheme is used to compute the spatial derivatives. The underlying curvilinear mesh near the vortex generator is designed properly with a large portion of the immersed boundary intersecting with the grid nodes to reduce the number of near-boundary nodes that require interpolation. The current immersed boundary technique was tested and verified on backward-facing step flow and flow past a circular cylinder. Numerical experiments have shown that the discrete direct forcing approach is very effective. In numerical simulation of flow behind the AVG, the computational results are comparable to the experimental data.

Numerical modeling of nonlinear water waves over heterogeneous porous beds

The transformation of nonlinear water waves over porous beds is studied by applying a numerical model based on Chen's [2006. Fully nonlinear Boussinesq-type equations for waves and currents over porous beds. Journal of Engineering Mechanics, 132:2, 220–230] Boussinesq-type equations for highly nonlinear waves on permeable beds. The numerical model uses a high-order time-marching solution and fourth-order finite-difference schemes for discretization of first-order spatial derivatives to obtain a computational accuracy consistent with the model equations. By forcing the wave celerity and spatial porous-damping rate of the linearized model to match the exact linear theory for horizontal porous bed over a prescribed range of relative depths, the values of the model parameters are optimally determined. Numerical simulations of the damped wave propagation over finite-thickness porous layer demonstrate the accuracy of both the numerical model and governing equations, which have been shown by prior theoretical analyses to be accurate for both nominal and thick porous layers. These simulations also elucidate on the significance of the higher-order porous-damping terms and the influence of the hydraulic parameters. Application of the model to the simulation of the wave field around a laboratory-scale submerged porous mound provides a measure of its capability, as well as useful insight into the scaling of the porous-resistance coefficients. For application to heterogeneous porous beds, the assumption of weak spatial variation of the porous resistance is examined using truncated forms of the governing equations. The results indicate that the complete set of Boussinesq-type equations is applicable to porous beds of nonhomogeneous makeup.

A simple form for the fourth order difference method for 3-D elliptic equations

In 1992, Jain et al. [M.K. Jain, R.K. Jain, R.K. Mohanty, Fourth-order finite difference method for three dimensional elliptic equations with nonlinear first-derivative terms, Numer. Meth. Part. Differ. Equat. 8 (1992) 575–591] proposed a fourth order finite difference scheme for the 3-D elliptic equation. In this paper, we present a simple and new form of 19-point fourth order difference method for the nonlinear second-order 3-D elliptic difference equation Au xx + Bu yy + Cu zz = f( x, y, z, u, u x , u y , u z ), where A, B and C are constants on a cubic region W subject to the Dirichlet boundary conditions on ∂ W.

A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation

A fourth-order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three-dimensional Poisson equation on a regular cubic domain. The difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three-dimensional manipulations. We use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth-order finite difference scheme.

Accuracy study of the IDO scheme by Fourier analysis

The numerical accuracy of the Interpolated Differential Operator (IDO) scheme is studied with Fourier analysis for the solutions of Partial Differential Equations (PDEs): advection, diffusion, and Poisson equations. The IDO scheme solves governing equations not only for physical variable but also for first-order spatial derivative. Spatial discretizations are based on Hermite interpolation functions with both of them. In the Fourier analysis for the IDO scheme, the Fourier coefficients of the physical variable and the first-order derivative are coupled by the equations derived from the governing equations. The analysis shows the IDO scheme resolves all the wavenumbers with higher accuracy than the fourth-order Finite Difference (FD) and Compact Difference (CD) schemes for advection equation. In particular, for high wavenumbers, the accuracy is superior to that of the sixth-order Combined Compact Difference (CCD) scheme. The diffusion and Poisson equations are also more accurately solved in comparison with the FD and CD schemes. These results show that the IDO scheme guarantees highly resolved solutions for all the terms of fluid flow equations.

Reduced-Order Model for Efficient Simulation of Synthetic Jet Actuators

A new reduced-order model of multidimensional synthetic jet actuators that combines the accuracy and conservation properties of full numerical simulation methods with the efficiency of simplified zero-order models is proposed. The multidimensional actuator is simulated by solving the time-dependent compressible quasi-1-D Euler equations, while the diaphragm is modeled as a moving boundary. The governing equations are approximated with a fourth-order finite difference scheme on a moving mesh such that one of the mesh boundaries coincides with the diaphragm. The reduced-order model of the actuator has several advantages. In contrast to the 3-D models, this approach provides conservation of mass, momentum, and energy. Furthermore, the new method is computationally much more efficient than the multidimensional Navier-Stokes simulation of the actuator cavity flow, while providing practically the same accuracy in the exterior flowfield. The most distinctive feature of the present model is its ability to predict the resonance characteristics of synthetic jet actuators; this is not practical when using the 3-D models because of the computational cost involved. Numerical results demonstrating the accuracy of the new reduced-order model and its limitations are presented.

Wave propagation using bases for bandlimited functions

We develop a two-dimensional solver for the acoustic wave equation with spatially varying coefficients. In what is a new approach, we use a basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a first-order system, we evolve the equation in time using the matrix exponential. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of one-dimensional operators. We also use a partitioned low-rank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and computational time when compared with a fourth-order finite difference scheme in space and an explicit fourth-order Runge–Kutta solver in time.

A fourth order finite difference method applied to elastodynamics: Finite element and boundary element formulations

This work presents a direct integration scheme, based on a fourth order finite difference approach, for elastodynamics. The proposed scheme was chosen as an alternative for attenuating the errors due to the use of the central difference method, mainly when the time-step length approaches the critical time-step. In addition to eliminating the spurious numerical oscillations, the fourth order finite difference scheme keeps the advantages of the central difference method: reduced computer storage and no requirement of factorisation of the effective stiffness matrix in the step-by-step solution. A study concerning the stability of the fourth order finite difference scheme is presented. The Finite Element Method and the Boundary Element Method are employed to solve elastodynamic problems. In order to verify the accuracy of the proposed scheme, two examples are presented and discussed at the end of this work.

Fourth‐order finite difference scheme for the incompressible Navier–Stokes equations in a disk

AbstractWe develop an efficient fourth‐order finite difference method for solving the incompressible Navier–Stokes equations in the vorticity‐stream function formulation on a disk. We use the fourth‐order Runge–Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth‐order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright © 2003 John Wiley & Sons, Ltd.

On the stability of a Bingham fluid flow in an annular channel

Linear stability of a fully developed Bingham fluid flow between two coaxial cylinders subject to infinitesimal axisymetric perturbations is investigated. The analysis leads to two uncoupled Orr–Sommerfeld equations with appropriate boundary conditions. The numerical solution is obtained using fourth order finite difference scheme. The computations were performed for various plug flow dimensions and radii ratios. Within the range of the parameters considered in this paper, the Poiseuille flow of Bingham fluid is found to be linearly stable. To cite this article: N. Kabouya, C. Nouar, C. R. Mecanique 331 (2003).

Free-surface turbulent wake behind towed ship models: experimental measurements, stability analyses and direct numerical simulations

We combine experimental, theoretical and numerical efforts to investigate the turbulent wake far behind a surface ship at model scales. Experimental measurements using digital particle image velocimetry (DPIV) are performed for the wakes of three towed hulls with beam-to-draught ratios b/d = 1, 2, 6. Based on model speed and beam, the Reynolds and Froude numbers are O(103) and O(10−2) respectively. Distinct surface features associated with persistent surface-normal vorticity have been identified, which are characterized by large-scale meandering structures. Both lateral and longitudinal scales of the meandering are quantified, with the former found to increase as b/d decreases and the latter independent of b/d. Based on measurements at multiple horizontal and vertical planes, profiles of the mean flow and fluctuation intensity for each velocity component are obtained. To understand the turbulence transition mechanism, an Orr–Sommerfeld stability analysis (OS) is formulated for the wake flow with free-surface boundary conditions, and solved by using a fourth-order finite-difference scheme. Unstable modes antisymmetric to the wake centre-plane are identified. Consistent with the experimental results, the growth rates of unstable modes increase substantially as b/d decreases, while the dependence of meandering wavelengths on b/d is found to be weak. Finally, we perform direct numerical simulation (DNS) of Navier–Stokes equations for the wake flow. The growth rates of unstable modes agree well with the predictions by OS analysis. Compared with experiments, DNS accurately captures the surface-normal vorticity signatures, the meandering features, as well as statistics of turbulence intensity. We also obtain from DNS a detailed description of enstrophy, turbulence length scales, and vortex structures for the wake flow.

A Simple Compact Fourth-Order Poisson Solver on Polar Geometry

We present a simple and efficient compact fourth-order Poisson solver in polar coordinates. This solver relies on the truncated Fourier series expansion, where the differential equations of the Fourier coefficients are solved by the compact fourth-order finite difference scheme. By shifting a grid a half mesh away from the origin and incorporating the symmetry constraint of Fourier coefficients, we can easily handle coordinate singularities without pole conditions. The numerical evidence confirms fourth-order accuracy for the problem on an annulus and third-order accuracy for the problem on a disk. In addition, a simple and comparably accurate approximation for the derivatives of the solution is also presented.

The Convergence of Spectral and Finite Difference Methods for Initial-Boundary Value Problems

The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high-order methods. Explicit application of boundary conditions (BCs) that are independent of the initial condition (IC) results in the compatibility conditions not being satisfied. Since this is the case in most science and engineering applications, it is shown that not only does the error in a spectral method, as measured in the maximum norm, converge algebraically, but the accuracy of finite differences is also reduced. For the heat equation with a parabolic IC and Dirichlet BCs, we prove that the Fourier method converges quadratically in the neighborhood of t=0 and the boundaries and quartically for large t when the first-order compatibility conditions are violated. For the same problem, the Chebyshev method initially yields quartic convergence and exponential convergence for t>0. In contrast, the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t. These results naturally direct attention to finite difference methods that are also algebraically convergent. In the case of the wave equation, we prove that a second-order finite difference method is reduced to 4/3-order convergence and numerically show that a fourth-order finite difference scheme is apparently reduced to 3/2-order. Finally, for the wave equation subject to general ICs and zero BCs, we give a conjectureon the error for a second-order finite difference scheme, showing that an O(N-2 log N) convergence is possible.

Preconditioned iterative methods for the nine-point approximation to the convection–diffusion equation

Iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth-order finite difference schemes for convection–diffusion problems. Simple recurrences for implementing the ILU(0) factorization of the nine-point scheme are derived. Different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices. In terms of algebraic properties of the preconditioned matrices, our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses. Comparison of the convergence rates of GMRES applied to the preconditioned linear systems is done with respect to the field of values, Ritz and harmonic Ritz values of the preconditioned matrices. Numerical results show that the GMRES residual norm decreases rapidly when the difference between the Ritz and harmonic Ritz values becomes small. We also describe the results of experiments when some well-known Krylov subspace methods are used to solve the linear system arising from the compact fourth-order discretizations.

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.

A new finite difference discretization of order four for for two-dimensional quasi-linear elliptic boundary value problem

We report a new algorithm of O(h 4) for the numerical solution of ux and uy for the solution of two dimensional quasi-linear elliptic equation subject to the Dirichlet boundary conditions. The proposed method requires only nine grid points on a uniform square grid and applicable to the problems both in cartesian and polar coordinates. We also discuss two sets of fourth-order finite difference methods; one in the absence of mixed derivative term, second when the coefficient of uxy is not equal to zero and the coefficients of uxx and u yy are equal. There do not exist fourth order finite difference schemes involving nine grid points for the general case. Numerical examples are given to illustrate the method and its fourth order convergence.

Technical note: A fourth-order finite difference scheme for a system of a 2D nonlinear elliptic partial differential equations

In this article we present a fourth-order finite difference scheme, for a system of two-dimensional, second-order, nonlinear elliptic partial differential equations with mixed spatial derivative terms, using 13-point stencils with a uniform mesh size h on a square region R subject to Dirichlet boundary conditions. The scheme of order h4 is derived using the local solution of the system on a single stencil. The resulting system of algebraic equations can be solved by iterative methods. The difference scheme can be easily modified to obtain formulae for grid points near the boundary. Computational results are given to demonstrate the performance of the scheme on some problems including Navier-Stokes equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 43–53, 2001

A new fourth order finite difference scheme for the heat equation

We propose and investigate a new simple fourth order finite difference scheme for the heat equation. Numerical simulations do confirm theoretical analysis of accuracy and stability condition.