Sixth-order Compact Finite Difference Scheme Research Articles

A high-order finite-difference solver for direct numerical simulations of magnetohydrodynamic turbulence

This paper presents the development and validation of a Magnetohydrodynamics (MHD) module integrated into the Xcompact3d framework, an open-source high-order finite-difference suite of solvers designed to study turbulent flows on supercomputers. Leveraging the Fast Fourier Transform library already implemented in Xcompact3d, alongside sixth-order compact finite-difference schemes and a direct spectral Poisson solver, both the induction and potential-based MHD equations can be efficiently solved at scale on CPU-based supercomputers for fluids with strong and weak magnetic field, respectively. Validation of the MHD solver is conducted against established benchmarks, including Orszag-Tang vortex and MHD channel flows, demonstrating the module's capability to accurately capture complex MHD phenomena, providing a powerful tool for research in both engineering and astrophysics. The scalability of the Xcompact3d framework remains intact with the incorporation of the MHD module, ensuring efficient performance on modern high-performance clusters. This paper also presents new findings on the evolution of the Taylor-Green vortex under an external magnetic field for different flow regimes.

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.

Two different temporal domain integration schemes combined with compact finite difference method to solve modified Burgers’ equation

The modified Burgers’ equation has application in various fields such as explosions and sonic boom theory, propagation of torsional waves in thin circular viscoelastic rod, acoustic waves produced by laser radiations, etc. A sixth-order compact finite difference scheme has been used for the space integration. Two different approaches have been followed for time integration. One is the Crank-Nicolson scheme in which the nonlinear term is handled with a quasilinearization process. Another is the strong stability preserving Runge-Kutta method of four stage, third-order convergence (SSP-RK43), in which no linearization is required. The stability analysis for both the schemes has been presented and is shown to be stable. Few test problems have been solved, demonstrating the performance of the proposed techniques. The L2 and L∞ error norms are calculated and are compared with the previous work. The aim of the paper is to solve such an important equation with better accuracy and less computational efforts.

Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions

Compact finite difference methods are very popular for solving differential equations that arise in a wide variety of real-world applications. Despite their popularity, the efficiency of these methods is limited by the need for matrix inversion which is troubling when the size of the matrix is very large. However, there is a growing demand for methods with better accuracy and computational speed. To meet such needs, we design a new method which uses a sixth-order compact finite difference scheme and a discrete sine transform to solve Poisson equations with Dirichlet boundary conditions. The scheme is developed using the concept of higher-order Taylor series expansion which is then used to discretize Poisson equations that results in a system of linear algebraic equations. To solve this system, we propose a discrete sine transform designed based on the developed compact finite difference scheme. We proved analytically and numerically that the order of convergence of the proposed method is six. The efficiency of the new scheme is demonstrated by solving different test problems. The numerical results indicate that the proposed method outperforms an existing fourth-order scheme and provides solutions that are in excellent agreement with the exact solution.

Numerical study on spatial scale characteristics of sound scattering by a static isentropic vortex

The scattering of acoustic waves by a vortex is a fundamental problem of the acoustic waves propagation in complex flow field, which plays an important role in academic research and engineering application for sound source localization, acoustic target recognition and detection, the far field noise prediction, such as aircraft wake vortex identification, detection and ranging, acoustic target forecasting in turbulent shear flow, acoustic measurement and sound source localization in wind tunnel test, etc. The nonlinear scattering phenomenon occurs when acoustic wave passes through the vortex, which is mainly related to the length-scale ratio between the wavelength of acoustic wave and the core radius of the vortex. In this paper, a plane acoustic wave passing through a stationary isentropic vortex is numerically simulated by solving a two-dimensional compressible, unsteady Euler equation. A sixth-order linear compact finite difference scheme is employed for spatial discretization. Time integration is performed by a four-stage fourth-order Runge-Kutta method. The eighth-order spatial compact filter scheme is adopted to suppress high frequency errors. At the far field boundaries, buffer layer is applied to handle the outgoing acoustic wave. Under the matching condition, the accuracy of the numerical results is verified by comparing with the previous direct numerical simulation results. The acoustic scattering cross-section method is introduced to analyze the effects of different length-scale ratio on the acoustic pulsation pressure, acoustic scattering effective sound pressure and acoustic scattering energy. Scattering occurs when sound waves pass through the vortex, the acoustic field in front of the vortex is basically unaffected, and the acoustic wave front remains intact. A “vacuum” region is formed slightly below the acoustic field directly behind the vortex, and two primary interference bands and several secondary interference bands are formed on the upper and lower sides of the vortex. As the length-scale ratio increases, the sound scattering decreases and the influence of the vortex flow field on the acoustic field gradually weakens. The influence region of effective sound pressure of acoustic scattering is mainly concentrated behind the vortex. With the increase of the length scale ratio, the influence gradually increases and extends to the upstream, and then the influence region gradually decreases to the vicinity of the vortex. When the length scale ratio is greater than or equal to 6, the location of the maximum effective sound pressure of sound scattering jumps from the upper right to the lower right of the vortex. The influence of acoustic wave wavelength change on the acoustic scattering energy can be divided into three parts. With the increase of the length scale ratio, the maximum sound scattering energy presents four different stages.

Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.

NUMERICAL STUDY OF VARIATION IN DRAG AND VIRTUAL MASS FORCES FOR A NANOFLUID FLOW THROUGH A MICROCHANNEL USING EULERIAN-EULERIAN TWO-PHASE MODEL

Laminar forced convection of copper oxide (CuO) and titanium oxide (TiO2 ) water-based nanofluid flow through a microchannel has been studied numerically using a Eulerian–Eulerian two-phase model. The governing equations of the liquid phase and solid phase are solved using sixth-order compact finite difference schemes. The effects of Reynolds number and nanoparticle volume concentrations on drag and virtual mass forces are reported. Estimation of thermal conductivity for nanofluid has been discussed here using two-phase results. Results show that significant heat transfer enhancement is observed with an increase in Reynolds number and nanoparticle volume concentrations compared with pure water. The 2 vol % CuO nanofluid has shown 3.44% and 18.65% enhancement in average Nusselt number compared to pure water for Re = 50 and Re = 600, respectively. An increase in particle concentration from 1% to 3% leads to a 4.45% increase in the average Nusselt number at Re = 200 for CuO nanofluid. A significant change of 26.5% is found in the dimensionless drag force with an increase in Reynolds number from 50 to 600. A negligible change in dimensionless drag force is observed with an increase in nanoparticle concentrations for Re ≥ 200. The virtual mass force is less significant than the drag force for nanofluid flows. Present results show a small deviation between the obtained and experimental thermal conductivity of nanofluid. The developed two-phase numerical model is validated with the numerical and experimental results available in the literature.

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

This paper modifies a n-dimensional Hopf-Cole transformation to the n-dimensional Burgers’ system. We obtain the n-dimensional heat conduction equation through the modification of the Hopf-Cole transformation. Then the fourth-order precise integration method (PIM) in combination with a spatially global sixth-order compact finite difference (CFD) scheme is presented to solve the equation with high accuracy. Moreover, coupling with the Strang splitting method, the scheme is extended to multi-dimensional (two, three-dimensional) Burgers’ system. Numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method. Moreover, the two-dimensional and three-dimensional examples demonstrate excellent adaptability, and the numerical simulation results also have very high accuracy in medium Reynolds numbers.

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4,4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4,4)-Padé approximation.

A Numerical Study on Fluid Flow and Acoustic Characteristics of a Supersonic Impinging Jet Using Vorticity Confinement

The objective of this work is to numerically study the fluid flow and acoustic field of a supersonic impinging jet by applying the vorticity confinement (VC) method. For this aim, the three-dimensional compressible Navier-Stokes equations with the incorporation of the VC method are considered and the resulting system of equations is solved by using the sixth-order compact finite-difference scheme. To eliminate the numerical instability, a low-pass high-order filter is used. The nonreflective boundary conditions are applied for all the free boundaries and the radiated sound field is obtained by the Kirchhoff surface integration. Comparisons of the present results with the experimental data and other numerical simulations show that the solution methodology adopted based on the application of the VC method with the high-order compact finite-difference scheme provides a good prediction of the fluid flow and the acoustic field of the impingement region on coarser grids than that usually required in the LESs, and thus, the calculations of coarse grid LESs are improved.

An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

In this paper, the combination of efficient sixth-order compact finite difference scheme (E-CFDS6) based proper orthogonal decomposition and Strang splitting method (E-CFDS6-SSM) is constructed for the numerical solution of the multi-dimensional parabolic equation (MDPE). For this purpose, we first develop the CFDS6 to attain a high accuracy for the one-dimensional parabolic equation (ODPE). Then, by the Strang splitting method, we have converted the MDPE into a series of one-dimensional ODPEs successfully, which is easier to implement and program with CFDS6 than an alternating direction implicit method. Finally, we employ proper orthogonal decomposition techniques to improve the computational efficiency of CFDS6 and build the E-CFDS6-SSM with fewer unknowns and sufficiently high accuracy. Six numerical examples are presented to demonstrate that the E-CFDS6-SSM not only can largely alleviate the computational load but also hold a high accuracy and simplify the process of the program for the numerical solution of MDPE.

Numerical investigation of the effect of airfoil thickness on onset of dynamic stall

Effect of airfoil thickness on onset of dynamic stall is investigated using large eddy simulations at chord-based Reynolds number of 200 000. Four symmetric NACA airfoils of thickness-to-chord ratios of 9 %, 12 %, 15 % and 18 % are studied. The three-dimensional Navier–Stokes solver, FDL3DI is used with a sixth-order compact finite difference scheme for spatial discretization, second-order implicit time integration and discriminating filters to remove unresolved wavenumbers. A constant-rate pitch-up manoeuver is studied with the pitching axis located at the airfoil quarter chord. Simulations are performed in two steps. In the first step, the airfoil is kept static at a prescribed angle of attack ($=4^{\circ }$). In the second step, a ramp function is used to smoothly increase the pitch rate from zero to the selected value and then the pitch rate is held constant until the angle of attack goes past the lift-stall point. The solver is verified against experiments for flow over a static NACA 0012 airfoil. Static simulation results of all airfoil geometries are also compared against XFOIL predictions with a generally favourable agreement. FDL3DI predicts two-stage transition for thin airfoils (9 % and 12 %), which is not observed in the XFOIL results. The dynamic simulations show that the onset of dynamic stall is marked by the bursting of the laminar separation bubble (LSB) in all the cases. However, for the thickest airfoil tested, the reverse flow region spreads over most of the airfoil and reaches the LSB location immediately before the LSB bursts and dynamic stall begins, suggesting that the stall could be triggered by the separated turbulent boundary layer. The results suggest that the boundary between different classifications of dynamic stall, particularly leading edge stall versus trailing edge stall, is blurred. The dynamic-stall onset mechanism changes gradually from one to the other with a gradual change in some parameters, in this case, airfoil thickness.

A reduced high-order compact finite difference scheme based on proper orthogonal decomposition technique for KdV equation

In this paper, a reduced implicit sixth-order compact finite difference (CFD6) scheme which combines proper orthogonal decomposition (POD) technique and high-order compact finite difference scheme is presented for numerical solution of the Korteweg-de Vries (KdV) equation. High-order compact finite difference scheme is applied to attain high accuracy for KdV equation and the POD technique is used to improve the computational efficiency of the high-order compact finite difference scheme. This method is validated by considering the simulation of five examples, and the numerical results demonstrate that the reduced sixth-order compact finite difference (R-CFD6) scheme can largely improve the computational efficiency without a significant loss in accuracy for solving KdV equation.

Algorithm 986

A collection of Matlab routines that compute derivative approximations of arbitrary functions using high-order compact finite difference schemes is presented. Tenth-order accurate compact finite difference schemes for first and second derivative approximations and sixth-order accurate compact finite difference schemes for third and fourth derivative approximations are discussed for the functions with periodic boundary conditions. Fourier analysis of compact finite difference schemes is explained, and it is observed that compact finite difference schemes have better resolution characteristics when compared to classical finite difference schemes. Compact finite difference schemes for the functions with Dirichlet and Neumann boundary conditions are also discussed. Moreover, compact finite difference schemes for partial derivative approximations of functions in two variables are also given. For each case a Matlab routine is provided to compute the differentiation matrix and results are validated using the test functions.

How does a high density ratio affect the near- and intermediate-field of high-Re hydrogen jets?

A comprehensive investigation of the mixing properties, in the near- and intermediate-field, of fully turbulent hydrogen round jets with high density ratios has been carried out by using an in-house 3D Large Eddy Simulation model. The model employs a non dissipative sixth-order compact finite difference scheme for space discretization and a fourth-order Runge-Kutta scheme for time integration. A Localized Artificial Diffusivity model has been used both to account for unresolved sub-grid scales and to avoid numerical instabilities. The model has been validated by comparing both the centerline mean velocity properties and the turbulence statistics of a low-Mach air into air jet with experimental results. A qualitative view of the jet coherent structures and a quantitative analysis of the turbulence scales have also been provided. The model has been used to investigate the structure of two low-Mach number hydrogen jets with the same momentum flux and very high ambient to jet density ratios, up to 50, and high Reynolds number, up to about 105. As expected, the centerline velocity decay rate is much larger than that of the air jet, however in the near- and intermediate-field the use of the classical definition of the effective diameter fails to collapse the velocity profiles onto a single slope and a recent definition of an effective diameter given by Sautet and Stepowski is required. Furthermore, the spreading rates of the two hydrogen jets are less affected by the density ratio with respect to the velocity decay rate.

A hybrid scheme for compressible magnetohydrodynamic turbulence

An efficient, high-resolution and oscillation-free hybrid scheme for shock-turbulence interactions in compressible magnetohydrodynamic (MHD) problems is presented. The hybrid scheme couples a sixth-order compact finite difference scheme in the smooth regions with a fifth-order WENO scheme in the shock regions. An eighth-order pentadiagonal filter is derived and utilized to maintain numerical stability and eliminate spurious oscillations. Various numerical examples are presented and the hybrid algorithm is proved to be accurate for smooth solutions and be able to capture discontinuities robustly. We have also shown the good capability of the hybrid scheme for the compressible MHD turbulence.

Comparison of Two Sixth-Order Compact Finite Difference Schemes for Burgers' Equation

In this study, two sixth-order compact finite difference schemes have been considered for solving the Burgers equation. The main difference of these schemes lies in the calculation of second-order derivative terms, which is obtained by applying the first-order operator twice and the method of undetermined coefficients. The aim is to comparison these schemes in terms of computational accuracy for solving the Burgers equation with difference viscosity values, especially for very small viscosity values. The results show that both schemes achieve almost the same accuracy for large viscosity values and second method is more accurate for moderate viscosity values, but both schemes are failed for very small viscosity values. However, when both schemes coupled low-pass filter for very small viscosity values, both schemes can well inhibit the problem.

Comparison of Two Sixth-Order Compact Finite Difference Schemes for Burgers' Equation

Comparison of Two Sixth-Order Compact Finite Difference Schemes for Burgers' Equation

A spectral-like turbulence-resolving scheme for fine sediment transport in the bottom boundary layer

A hybrid spectral-compact finite difference scheme for turbulent-resolving simulation of fine sediment transport in the bottom boundary layer is presented. The numerical approach extends an earlier pseudo-spectral model for direct numerical simulation (DNS) of turbulent flows with a sixth-order compact finite difference scheme in the wall-normal direction on Chebyshev grid points. The compact finite difference scheme allows easy implementation of flow-dependent properties (e.g., viscosity, diffusivity and settling velocity) and more flexible boundary conditions while still maintain spectral-like numerical accuracy. The numerical model is verified with analytical solutions of flow velocity and particle concentration of two simple Newtonian rheological closures under laminar condition. Prior laboratory and DNS data of turbulent channel flow are also used to validate the code. Several numerical simulations were carried out in a turbulent channel flow setting to investigate the interplay between the two turbulence modulation mechanisms induced by the presence of sediment, namely sediment-induced density stratification and enhanced viscosity due to rheological stress. We demonstrate that at the Reynolds number, Richardson number, and non-dimensional settling velocity used here, the flow remains turbulent but sediment-induced density stratification already causes noticeable damping of turbulence (drag reduction). By further introducing a Newtonian rheological stress into the system, flow turbulence is further damped by the increased effective viscosity, which can trigger laminarization.

A sixth-order fast algorithm for the electromagnetic scattering from large open cavities

In this paper, a sixth-order fast algorithm is presented for solving the electromagnetic scattering problem from a rectangular open cavity embedded in an infinite ground plane. Firstly, a sixth-order compact finite difference scheme is employed to discretize the scattering problem described by the Helmholtz equation with a nonlocal boundary condition. Then, exploiting the discrete Fourier transformation in the horizontal and a Gaussian elimination in the vertical direction presented by [2], the discrete system reduces to a much smaller interface system. An effective preconditioner is developed for the Krylov subspace iterative solver to solve this interface system. Numerical results show the remarkable efficiency and accuracy of the new algorithm.