Sixth-order Compact Finite Difference Research Articles

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

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.

A Comprehensive Comparison in the Heat Transfer Performance of Pure Water-Based and Liquid Gallium-Based Hybrid Nanofluid Flows through a Minichannel, Using Two-Phase Eulerian–Eulerian Model

An in-house Fortran-based two-phase Eulerian–Eulerian solver is used to numerically study water-based and liquid metal gallium-based alumina-copper hybrid nanofluid forced convection flow through a wide rectangular mini channel in the laminar regime. The governing equations of liquid and solid phases are solved using sixth-order compact finite difference method. A parametric study involving Reynolds number, particle concentration and mixture ratios has been reported. From the study it is found that pure liquid gallium has 43.5% greater heat transfer enhancement compared to pure water. The ‘heat transfer performance parameter’ is used to analyze the heat transfer characteristics of all the nanofluids considered. Addition of copper nanoparticles into base fluid improves the heat transfer performance of liquid gallium by 10.73% and pure water by 46.9%, at 1% particle concentration and a Reynolds number of 100. A significant change in heat transfer performance is found with a change in the Reynolds number and particle concentration for water-based hybrid nanofluids compared to liquid gallium-based hybrid nanofluids. A Nusselt number correlation is proposed for alumina-copper water-based hybrid nanofluids. Present results are validated with numerical and experimental results available in the literature.

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.

Numerical Approximation of Generalized Burger’s-Fisher and Generalized Burger’s-Huxley Equation by Compact Finite Difference Method

In this work, computational analysis of generalized Burger’s-Fisher and generalized Burger’s-Huxley equation is carried out using the sixth-order compact finite difference method. This technique deals with the nonstandard discretization of the spatial derivatives and optimized time integration using the strong stability-preserving Runge-Kutta method. This scheme inculcates four stages and third-order accuracy in the time domain. The stability analysis is discussed using eigenvalues of the coefficient matrix. Several examples are discussed for their approximate solution, and comparisons are made to show the efficiency and accuracy of CFDM6 with the results available in the literature. It is found that the present method is easy to implement with less computational effort and is highly accurate also.

Sixth order compact finite difference schemes for Poisson interface problems with singular sources

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem −∇2u=f in Ω∖Γ with Dirichlet boundary condition such that f is smooth in Ω∖Γ and the jump functions [u] and [∇u⋅n→] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of −∇2u=f+gδΓ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇u⋅n→ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω+ and Ω−. The coefficient matrix A in the resulting linear system Ax=b, following from the proposed scheme, is independent of any source term f, jump condition gδΓ, interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω+ or Ω−. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

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.

An efficient high-order compact finite difference method for the Helmholtz equation

This paper is devoted to applying the sixth-order compact finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.

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 new development of sixth order accurate compact scheme for the Helmholtz equation

A standard sixth order compact finite difference scheme for two dimensional Helmholtz equation is further improved and a more accurate scheme is presented for the two dimensional Helmholtz equation which is also compact. The novel feature of the present scheme is that it is less sensitive to the associated wave number when compared with that for available sixth order schemes. Theoretical analysis is presented for the newly constructed scheme. The high accuracy of the proposed scheme is illustrated by comparing numerical solutions for solving the two dimensional Helmholtz equations using available sixth-order schemes and the present scheme.

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.

Effect of the nodes near boundary points on the stability analysis of sixth-order compact finite difference ADI scheme for the two-dimensional time fractional diffusion-wave equation

In this paper, the aim is to present a high-order compact alternating direction implicit (ADI) scheme for the two-dimensional time fractional diffusion-wave (FDW) equation. The time fractional derivative which has been described in the Caputo’s sense is approximated by a scheme of order O(τ3−α), 1<α<2 and the space derivatives are discretized with a sixth-order compact procedure. The solvability, stability and H1 norm of the scheme are proved. Numerical results are provided to verify the accuracy and efficiency of the proposed method of solution. The sixth-order accuracy in the space directions has not been achieved in previously studied schemes.

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.