High-order Compact Scheme Research Articles

Combined high order compact schemes for non-self-adjoint nonlinear Schrödinger equations

Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives ux‾, uxx in the NSANLSE. If one uses classical high order compact schemes to approximate uxx and ux‾ separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves uxjn and uxxjn simultaneously in terms of uj. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.

Two-level implicit high-order compact scheme in exponential form for 3D quasi-linear parabolic equations

We discuss a new high accuracy compact exponential scheme of order four in space and two in time to solve the three-dimensional quasi-linear parabolic partial differential equations. The derived half-step discretization based scheme is implicit in nature and demands only two levels for computation. The generalization of the proposed exponential scheme for the system of the quasi-linear parabolic PDEs is also represented. We generate unconditionally stable alternating direction implicit scheme for the linear parabolic equation in general form. The accuracy and the theoretical results of the proposed scheme are verified for high Reynolds number by several numerical problems like linear and non-linear convection-diffusion equation, coupled Burgers' equations, Navier-Stokes equations, quasi-linear parabolic equation, etc.

High-order compact gas-kinetic scheme in arbitrary Lagrangian-Eulerian formulation

This study proposes an extension of the high-order compact gas-kinetic scheme (CGKS) to compressible flow simulation in an arbitrary Lagrangian-Eulerian (ALE) formulation in unstructured mesh. The ALE method is achieved by subdividing arbitrary mesh into tetrahedrons and integrating flux function in a local coordinate system at the cell interface to ensure geometric conservation law. The scheme incorporates a compact reconstruction with third-order accuracy for updating both cell-averaged conservative flow variables and their gradients. HWENO-type nonlinear reconstruction and gradient compression factors are adopted to improve the accuracy and robustness of the scheme. A multi-stage multi-derivative (MSMD) time-stepping method is also implemented to achieve high-order time accuracy with fewer middle stages. The scheme is used to study problems involving moving boundaries. The numerical experiments demonstrate the effectiveness of the scheme in capturing the accurate solutions of both low-speed smooth flow and highly compressible ones with strong shock waves.

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Generalized high-order compact difference schemes for the generalized Rosenau–Burgers equation

Flow dynamics and heat transfer in the wake of an rotary oscillating circular cylinder with an isothermal control plate

This research explores the flow dynamics and heat transfer within the wake of a rotary oscillating circular cylinder in the presence of an isothermal control plate. Two-dimensional, unsteady, viscous and laminar flow of a Newtonian fluid is considered using the Higher Order Compact Scheme (HOC) to discretize the governing equations and the Bi-Conjugate Gradient Stabilized method to solve the resulting system. Simulations are performed for various gap ratios, maximum angular velocities, and frequency ratios of oscillation at Prandtl number 0.7 and Reynolds number 150 using an in-house code. Results show a significant increase in heat transmission for d / R 0 = 0.5 and f / f 0 = 0.5 for all αm . Drag and lift coefficients are also analyzed, with the maximum peak of the drag coefficient decreasing by 9.88% for d / R 0 = 3 . This research offers valuable insights for advancing and optimizing aerodynamic forces and heat transfer processes, particularly in the field of fluid dynamics. The study focuses on enhancing the efficiency of heat transfer mechanisms around rotary oscillating circular cylinders, contributing to the development of cutting-edge technologies in aerodynamics and thermal management.

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A Positivity-Preserving and Well-Balanced High Order Compact Finite Difference Scheme for Shallow Water Equations

A third-order compact finite volume scheme on unstructured grid for fluid flows

The large reconstruction stencil is one of the biggest obstacles in design and implementation of high-order finite volume schemes on unstructured grid. To tackle this problem, a third-order compact reconstruction scheme on unstructured grid is developed on the basis of compact least-squares (CLS) finite volume method. In building the reconstruction polynomial, the general boundary condition is adopted to ensure the unified third-order accuracy in both the internal and boundary cells. The reconstruction polynomial is solved in a semi-implicit sense to ensure high computational efficiency without solving the global linear algebraic equations. The theoretical accuracy order and stability limit are firstly verified by the Fourier analysis. The numerical accuracy and efficiency are then verified and validated with several incompressible and compressible flow cases. This work provides a reference to design and analysis of high-order compact finite volume schemes on unstructured grid with unified accuracy.

Fourth-order compact block-centered splitting domain decomposition method for parabolic equations

It is of importance to develop efficient high-order solution-flux domain decomposition methods for solving the large scale time-dependent partial differential equations. In this paper, a fourth-order compact block-centered splitting domain decomposition method is developed for solving parabolic partial differential equations in large domains. Combining the time second-order splitting technique with non-overlapping domain decompositions, we propose the compact block-centered splitting domain decomposition algorithm to the solution-flux structured system of parabolic partial differential equations. The domain is divided into non-overlapping multi-block sub-domains of block-centered meshes. On the interfaces of sub-domains, the interface fluxes are predicted by the semi-implicit interface flux scheme while the solution and flux in the interiors of sub-domains are computed by the compact block-centered implicit solution-flux scheme. The feature of the method is that over block-centered grids, constructing high-order compact difference scheme to the structured system of solution and flux equations leads to efficient schemes for solving the problems over domain decompositions. Numerical experiments show that the method is of fourth-order convergence in space and of second order convergence in time and it is much efficient in parallel computing.

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Analysis of the Dominant Structures of an Impinging Round Jet at M=0.9

Large-eddy simulation of subsonic round jet impinging on a flat circular plate is performed at Mach number 0.9 and Reynolds number 25,000, based on the diameter and centerline velocity of the jet for two impingement distances h of 6r0 and 8r0 using high-order compact finite difference scheme. The complex flow phenomena associated with the impinging jet flowfield are studied. The power spectral density (PSD) of instantaneous pressure reveals a discrete impingement tone. Dynamic mode decomposition (DMD) of the velocity and pressure is also done for both cases to extract the most dominant modes of the flow. The fundamental impingement tone frequency obtained from the PSD of pressure is identical to that of the first dominant pressure mode obtained from DMD. Both axisymmetric and semihelical modes are present for the h=8r0 case, but only axisymmetric modes are there in the h=6r0 case, and it is noted that the modes corresponding to the fundamental tone have an axisymmetric structure. Also, one dominant mode showing the wall jet structures denoting the liftoff of fluid from the wall after impingement at some radial distance away from the stagnation region is present in both impingement distance cases.

Analysis of thermsolutal performance and entropy generation for ternary hybrid nanofluid in a partially heated wavy porous cabinet

PurposeThis study comprehensively examines entropy generation and thermosolutal performance of a ternary hybrid nanofluid in a partially active porous cabinet. The purpose of this study is to comprehend the intricate phenomena of double diffusion by investigating the dispersion behavior of Al2O3, CuO, and Ag nanoparticles in water.Design/methodology/approachThe cabinet design consists of two horizontal walls and two curved walls with the lower border divided into a heated and concentrated region of length b and the remaining sections are adiabatic. The vertical borders are cold and low concentration, while the upper border is adiabatic. Two cavity configurations such as convex and concave are considered. A uniform porous medium is taken within the ternary hybrid nanofluid. This has been characterized by the Brinkman-extended Darcy model. Thermosolutal phenomena are governed by the Navier-Stokes equations and are solved by adopting a higher-order compact scheme.FindingsThe present study focuses on exploring the influence of several well-defined parameters, including Rayleigh number, Darcy number, Lewis number, Buoyancy ratio number, nanoparticle volume concentration and heater size. The results indicate that the ternary hybrid nanofluid outperforms both the mono and hybrid nanofluids in all considered aspects.Originality/valueThis study brings forth a significant contribution by uncovering novel flow features that have previously remained unexplored. By addressing a well-defined problem, the work provides valuable insights into the enhancement of thermal transport, with direct implications for diverse engineering devices such as solar collectors, heat exchangers and microelectronics.

High-order compact gas-kinetic scheme for two-layer shallow water equations on unstructured mesh

For the two-layer shallow water equations, a high-order compact gas-kinetic scheme (GKS) on triangular mesh is proposed. The two-layer shallow water equations have complex source terms in comparison with the single layer equations. The main focus of this study is to construct a time-accurate evolution solution at a cell interface and to design a well-balanced scheme. The evolution model at a cell interface provides not only the numerical fluxes, but also the flow variables. In the development of the well-balanced scheme, the time-dependent flow variables at the cell interfaces surrounding the cell can be utilized to update the cell-averaged gradients. These time-evolving gradients are subsequently applied in the discretization of source terms inside each control volume. Based on the cell-averaged flow variables and their gradients, high-order initial data reconstruction can be achieved with compact stencils. The compact high-order GKS has advantages to simulate the flow evolution in complex domain covered by unstructured mesh. Many test cases are used to validate the accuracy and robustness of the scheme for the two-layer shallow water equations.

Numerical simulation of acoustic streaming in standing waves

This study presents a numerical investigation of acoustic streaming motion (of the Rayleigh type) in a compressible gas inside two-dimensional rectangular enclosures. To numerically study the effects of the sound field intensity on the formation process of streaming structures, we propose to discretize the fully compressible form of the two-dimensional Navier-Stokes equations using a high-order (formally greater or equal to fourth-order) accurate numerical scheme in both space and time. The proposed numerical solver utilizes high-order compact schemes along each spatial dimension combined with a filtering procedure when it is necessary. Acoustic standing waves are excited inside the enclosures and the resulting acoustic streaming patterns are investigated for low and high-intensity waves, in both linear and nonlinear regimes following closely the work of Daru et al. (2013). An extended investigation indicates that without the incorporation of an appropriate filter, the application of the high-order compact schemes in case of fast streaming results in spurious oscillations which inhibit their applicability. Following the recent relevant literature, the numerical simulations performed demonstrate the ability of the proposed numerical approach to reproduce efficiently and robustly the transitions from regular acoustic streaming to irregular streaming and the relevant phenomena confirming results that have been previously presented.

MHD natural convection flow in a porous medium-filled corrugated enclosure: Effect of heat sources with different heights

A two-dimensional natural convection flow problem in a porous medium-filled enclosure with electrically conducting fluids and solid heat sources of different heights is numerically investigated in this study. The top boundary is cold, and the vertical side boundaries are insulated, while rectangular-shaped corrugations at the lower boundary are generated by placing heat sources. Three distinctive cases are considered concerning the height of the solid rectangular-shaped heat sources. The nonlinear coupled transport equations are discretized by using a higher-order compact (HOC) finite difference scheme. First, existing numerical and experimental data are used to validate the developed code. Then, the effect of key parameter values, like Hartmann number (25–150), Rayleigh number (103–106), Darcy number (10−5–10−1), and Prandtl number (0.015–10), on the convective flow in the corrugated enclosure is analysed. The influence of the permeability of the porous medium is abundantly seen for a range of significant parameter values with a fixed high Rayleigh-Darcy number. The formation of primary, secondary, tertiary, and quaternary vortices affects the thickness of the thermal boundary layer on both hot and cold surfaces in both symmetric and asymmetric patterns. In addition, possible multiple steady states are observed in the Case 2 at Pr=10.0. The local and average Nusselt numbers remains constant on the bottom cold surfaces of the corrugated enclosure, while significant variations are observed on the upper surfaces of the heat sources. These findings are analysed by conducting stream-function (ψ), isotherms (T), local (Nuh,Nuv), and average (Nū) Nusselt number plots. The intensities of heat transfer rate are also examined by analysing Nusselt numbers as a function of Prandtl, Darcy, Rayleigh and Hartmann numbers. Analysing natural convective heat transfer fluid flow in an enclosure with heat sources is prominent for ample engineering applications such as solar collectors, designing nuclear reactors, chemical energy production systems, building constructions and microelectronic equipment, to name a few.

An efficient high-order compact approach for spiral wave dynamics by the FHN model

We simulate spiral waves through the famous Fitzhugh–Nagumo (FHN) model of excitable media by employing zero flux and periodic boundary conditions. This model has been solved numerically by modifying a recently developed implicit high-order compact (HOC) finite difference scheme. We also propose a novel approach for implementing the HOC scheme on boundaries of the domain for the periodic boundary conditions. Henceforth, we investigate the effects of the zero flux and periodic boundary conditions on the patterns in excitable media and the resulting dynamics. The introduction of periodic boundary conditions is seen to trigger the breakup of the spiral waves as opposed to the ones found by imposing no-flux boundary conditions. Our computed grid independent solutions are found to be extremely close to well-known numerical results. In the process, we also probe the unconditional stability of the HOC discretization for the FHN model.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd s systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation

A high-order compact finite difference scheme and its analysis for the time-fractional diffusion equation

Accurate derivatives approximations and applications to some elliptic PDEs using HOC methods

For many application problems that are modeled by partial differential equations (PDEs), not only it is important to obtain accurate approximations to the solutions, but also accurate approximations to the derivatives of the solutions. In this study, some new high order compact (HOC) finite difference schemes are derived to approximate the first and second derivatives of the solution to some elliptic PDEs using the numerical solution obtained from a HOC scheme applied to the same PDE. Convergence analysis for the computed derivatives is also presented to show that the order of the convergence is the same as that of the solution. The new HOC schemes for computing partial derivatives at both interior and boundary grid points take into account of the partial differential equations including the source term and/or the boundary conditions (Dirichlet, Neumann, or Robin). Fourth order accurate compact finite difference formulas with pre-computed coefficients and weights are developed for Poisson/Helmholtz PDEs, and code generated coefficients for diffusion-advection equations with constant coefficients. One important application is a new fourth-order compact scheme for solving incompressible Stokes equations with periodic boundary conditions.