Sixth-order Compact Difference Scheme Research Articles

Sixth-order compact difference scheme and multigrid method for solving the Poisson equation

Sixth-order compact difference scheme and multigrid method for solving the Poisson equation

Highly Efficacious Sixth-Order Compact Approach with Nonclassical Boundary Specifications for the Heat Equation

This paper suggests an accurate numerical method based on a sixth-order compact difference scheme and explicit fourth-order Runge–Kutta approach for the heat equation with nonclassical boundary conditions (NCBC). According to this approach, the partial differential equation which represents the heat equation is transformed into several ordinary differential equations. The system of ordinary differential equations that are dependent on time is then solved using a fourth-order Runge–Kutta method. This study deals with four test problems in order to provide evidence for the accuracy of the employed method. After that, a comparison is done between numerical solutions obtained by the proposed method and the analytical solutions as well as the numerical solutions available in the literature. The proposed technique yields more accurate results than the existing numerical methods.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

Efficiency of numerical schemes for two dimensional Gray Scott model

In this article, efficient numerical schemes for the two dimensional Gray Scott model are presented. The Gray Scott model presents self-replicating patterns such as spots and strips. These pattern formulations are suitable interplay between diffusion and reactions in which the coupled partial differential system is solved by using three finite difference schemes to enhance accuracy while maintaining stability of the system. The stability analysis is performed on stationary points whereas the analytical solution is compared with the numerical schemes, such as Douglas implicit fourth and sixth order compact difference schemes. The later two schemes are implemented for first time on such a system for analyzing error residuals and system efficiency. It is predicted that the efficiency is upgraded by Thomas block tridiagonal solver, which leads to an excellent improvement in accuracy measured by L∞ norm.

Cascadic multigrid methods combined with sixth order compact scheme for poisson equation

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.

A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation

This paper introduces a novel method for deducing high-order compact difference schemes for the two-dimensional (2D) Poisson equation. Like finite volume method, a dual partition is introduced. Combining Simpson integral formula and parabolic interpolation, a family of fourth-order and sixth-order compact difference schemes are obtained based on three different types of dual partitions. Moreover, several new fourth-order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth-order schemes which have been presented in others’ work. The outline for the nonlinear Poisson equation is also given. Numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth-order and sixth-order compact difference schemes.

Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes

A higher order compact difference (HOC) scheme with uniform mesh sizes in different coordinate directions is employed to discretize a two- and three-dimensional Helmholtz equation. In case of two dimension, the stencil is of 9 points while in three-dimensional case, the scheme has 27 points and has fourth- to fifth-order accuracy. Multigrid method using Gauss-Seidel relaxation is designed to solve the resulting sparse linear systems. Numerical experiments were conducted to test the accuracy of the sixth-order compact difference scheme with Multigrid method and to compare it with the standard second-order finite-difference scheme and fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme are established by using the errors normsl2.

Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation forPe≤5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified through the presented results and the literature. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the advection-diffusion equation. The present technique is seen to be a very reliable alternative to existing techniques for these kinds of applications.

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.

Numerical analysis of the immersed boundary method applied to the flow around a forced oscillating cylinder

In present paper, Navier-Stokes and Continuity equations for incompressible flow around an oscillating cylinder were numerically solved. Sixth order compact difference schemes were used to solve the spatial derivatives, while the time advance was carried out through second order Adams Bashforth accurate scheme. In order to represent the obstacle in the flow, the Immersed Boundary Method was adopted. In this method a force term is added to the Navier-Stokes equations representing the body. The simulations present results regarding the hydrodynamic coefficients and vortex wakes in agreement to experimental and numerical previous works and the physical lock-in phenomenon was identified. Comparing different methods to impose the IBM, it can be concluded that no alterations regarding the vortex shedding mode were observed. The Immersed Boundary Method techniques used here can represent the surface of an oscillating cylinder in the flow.

Computational and Experimental Investigation of a Nonslender Delta Wing

Computational simulations have been performed for a 50-deg-sweep delta wing with a sharp leading edge at a 15 deg angle of attack and moderate Reynolds numbers of Re = 2 x 10 5 , Re = 6.2 x 10 5 , and Re = 2 x 10 6 . A sixth-order compact-difference scheme with an eighth-order low-pass filter is used to solve the Navier-Stokes equations. Turbulence modeling has been accomplished using an implicit large eddy simulation method that exploits the high-order accuracy of the compact-difference scheme and uses the discriminating higher-order filter to regularize the solution. Computations have been performed on a baseline mesh of 11.3 x 10 6 grid points and a refined mesh of 35 x 10 6 grid points. An assessment of grid resolution showed that significantly finer-scale features of the flow could be captured on the refined mesh, providing a more accurate representation of the complex, unsteady, separated flow. Comparisons are also made with high-resolution particle image velocimetry images obtained for the two lower Reynolds numbers. The numerical results are examined to provide a description of the mean and instantaneous flow structure over the delta wing, including the separated vortical flow, vortex breakdown, surface flow features, and surface boundary-layer transition near the symmetry plane. The effect of Reynolds number on each of these features is assessed.

Computational study of shock interaction with a vortex ring

The problem of shock interaction with a vortex ring is investigated within the framework of axisymmetric Euler equations solved numerically by a shock-fitted sixth-order compact difference scheme. The vortex ring, which is based on Lamb’s formula, has an upstream circulation Γ=0.01 and its aspect ratio R lies in the range 8⩽R⩽100. The shock Mach number varies in the range 1.1⩽M1⩽1.8. The vortex ring/shock interaction results in the streamwise compression of the vortex core by a factor proportional to the ratio of the upstream and downstream mean velocity U1/U2, and the generation of a toroidal acoustic wave and entropy disturbances. The toroidal acoustic wave propagates and interacts with itself on the symmetry axis of the vortex ring. This self-interaction engenders high amplitude rarefaction/compression pressure peaks upstream/downstream of the transmitted vortex core. This results in a significant increase in centerline sound pressure levels, especially near the shock (due to the upstream movement of the rarefaction peak) and in the far downstream (due to the downstream movement of the compression peak). The magnitude of the compression peak increases nonlinearly with M1. For a given M1, vortex rings with smaller aspect ratios (R<20) generate pressure disturbances whose amplitudes scale inversely with R, while vortex rings with larger aspect ratios (R>40) generate pressure disturbances whose amplitudes are roughly independent of R.

Direct calculation of waves in fluid flows using high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expression for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations. The numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the Courant-Friedrichs-Lewy (CFL) limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about six - eight grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 - 300 time steps. Three numerical examples of waves in compressible flow are included: (1) sound propagation in a duct with linear shear, (2) linear wave growth in a compressible free shear layer, and (3) vortex pairing in a compressible free shear layer perturbed at two frequencies.