High-order Finite Difference Scheme Research Articles

A new higher-order super compact finite difference scheme to study three-dimensional non-Newtonian flows

This work introduces a new higher-order accurate super compact (HOSC) finite difference scheme for solving complex unsteady three-dimensional (3D) non-Newtonian fluid flow problems. As per the author's knowledge, the proposed scheme is the first ever developed higher-order compact finite difference scheme to solve 3D non-Newtonian flow problems. The proposed scheme is fourth-order accurate in space and second-order accurate in time, utilizing only seven adjacent grid points at the (n+1)th time level for the finite difference discretization. A time-marching methodology is employed with pressure calculated via a pressure-correction strategy based on the modified artificial compressibility method. Using the power-law viscosity model, we tackle the benchmark problem of a 3D lid-driven cavity, systematically analyzing the varied rheological behavior of shear-thinning (n = 0.5), shear-thickening (n = 1.5), and Newtonian (n = 1.0) fluids across different Reynolds numbers (Re=1,50,100,200). Both Newtonian and non-Newtonian results are carefully investigated in terms of streamlines, velocity variation, pressure distributions, and viscosity contours, and the computed results are validated with the existing benchmark results. The findings demonstrate excellent agreement with the existing results. It is found that for shear-thinning fluid (n = 0.5), u velocity is higher near the top moving wall than the case of Newtonian (n = 1.0) and shear-thickening fluid (n = 1.5) for all Re values. This extensive analysis, using the new HOSC scheme, not only increases our understanding of non-Newtonian fluid behavior but also provides a robust foundation for future research and practical applications.

Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.

High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

This paper develops high-order accurate well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers M⩾2) shallow water equations (SWEs) with non-flat bottom topography on both fixed and adaptive moving meshes, extending our previous work on the single-layer shallow water magnetohydrodynamics [25] and single-layer SWEs on adaptive moving meshes [58]. To obtain an energy inequality, the convexity of an energy function for an arbitrary M is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into M identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit strong stability preserving Runge-Kutta methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [58]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.

Numerical treatment of singular ODEs using finite difference and collocation methods

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class.In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes.

A matrix-free parallel two-level deflation preconditioner for two-dimensional heterogeneous Helmholtz problems

We propose a matrix-free parallel two-level deflation method combined with the Complex Shifted Laplacian Preconditioner (CSLP) for two-dimensional heterogeneous Helmholtz problems encountered in seismic exploration, antennas, and medical imaging. These problems pose challenges in terms of accuracy and convergence due to scalability issues with numerical solvers. Motivated by the limitations imposed by excessive computational time and memory constraints when employing a sequential solver with constructed matrices, we parallelize the two-level deflation method without constructing any matrices. Our approach utilizes preconditioned Krylov subspace methods and approximates the CSLP preconditioner with a parallel geometric multigrid V-cycle. For the two-level deflation, standard inter-grid deflation vectors and further high-order deflation vectors are considered. As another main contribution, the matrix-free Galerkin coarsening approach and a novel re-discretization scheme as well as high-order finite-difference schemes on the coarse grid are studied to obtain wavenumber-independent convergence. The optimal settings for an efficient coarse-grid problem solver are investigated. Numerical experiments of model problems show that the wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

High order numerical treatment of the Riesz space fractional advection-dispersion equation via matrix transform method

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method.

Numerical study of magneto-hydrodynamics natural convective flow in a porous-corrugated enclosure using a higher-order compact finite difference scheme

This study explores the numerical investigation of two-dimensional natural convection in a porous corrugated enclosure with heat sources of different heights and electrically conducting fluids. The governing equations are solved using a higher-order compact finite difference scheme. Boundaries on the top and sides are subject to thermal cooling and adiabatic conditions, respectively. Numerical results are presented for wide range of Rayleigh (103 to 106), Darcy (10−5 to 10−1), Hartmann (25 to 150), and Prandtl (0.015 to 10) numbers. For fixed high Rayleigh-Darcy values, the porous medium has higher permeability. Most significant possible multiple steady solutions can be found in the case 1 at Ra=106,Da=10−2,Ha=50, and Pr=10.0. In the low to moderate critical values of Prandtl numbers, it encompasses a wide range of fluid flow phenomena, from stable to unstable. The maximum enhancement of the averaged Nusselt numbers on the boundaries in increasing and decreasing trends are viz. Ra(139.13% and 54.37%), Da(46.14% and 74.25%), Ha(633.04% and 74.73%), and Pr(142.52% and 45.92%) for case 1, Ra(129.64% and 0.32%), Da(40.68% and 69.66%), Ha(46.65% and 80.85%), and Pr(18.41% and 18.23%) for case 2, and Ra(163.98% and 77.20%), Da(8.25% and 58.68%), Ha(81.38% and 59.17%), and Pr(30.60% and 22.46%) for case 3.

Higher-order compact finite difference method for a free boundary problem of ductal carcinoma in situ

A higher-order compact finite difference scheme is established to investigate the model of ductal carcinoma in situ, a free boundary value problem. For the first time, to determine the growth of a tumor numerically, a compact finite difference method is applied in the spatial direction, and the Crank-Nicolson scheme is used in the temporal direction. Through stability analysis, it has been demonstrated that the proposed numerical scheme is unconditionally stable. The convergence of the numerical scheme is analyzed theoretically by von Neumann's method. It has been proved that the proposed numerical method is second-order convergent in time and fourth-order convergent in space in the L2-norm. Two numerical examples are provided to verify the scheme's efficiency and validate the theoretical results. The tabular and graphical representations confirm the high accuracy and adaptability of the scheme.

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

New higher-order accurate super-compact scheme for three-dimensional natural convection and entropy generation

Exploring natural convection in three dimensions through numerical analysis has become essential for gaining a deeper understanding of this process compared to studying it in just two dimensions. This paper presents an enhanced analysis of three-dimensional (3D) natural convection phenomena within an air-filled cubical cavity, with a primary focus on heat transfer characteristics. The distinguishing feature of this research lies in the pioneering application of the newly developed higher-order accurate super-compact finite difference scheme to study 3D natural convection and entropy generation. This numerical method is renowned for its fourth-order spatial accuracy and second-order temporal accuracy. The super-compact scheme employs 19 grid points at the current time level (nth time level) while utilizing just seven grid points from the subsequent time level [(n+1)th time level. The flow is considered to be 3D, time-varying, laminar, and incompressible. First, the newly developed numerical code has undergone validation through quantitative and qualitative comparisons with existing results. Then, we analyze the flow phenomena and heat transfer dynamics within the cavity for various Rayleigh numbers (102≤Ra≤105) with a fixed Prandtl number (Pr = 0.71) by using this scheme. Various parameters such as local Nusselt numbers, averaged Nusselt numbers, local entropy, Bejan number, streamlines, and isotherm dispersion are studied in this work. It is found that as the Rayleigh number (Ra) increases, the Bejan number (Be) decreases, while the total entropy increases. When the Ra is less than or equal to 104, the Be remains above 0.5, indicating that irreversibility primarily arises from heat transfer. However, as we transition to Ra=105, the Be falls below 0.5, signaling that irreversibility is now predominantly driven by viscous effects. To the best of our knowledge, this research marks a pioneering effort by introducing the application of the higher order super-compact (HOSC) scheme for the examination of heat transfer within a 3D cubic cavity, thus endowing our work with a truly novel and pioneering character. It is worth mentioning that, before this study, there has been no precedent in the exploration of heat transfer using the super-compact scheme, thus endowing our work with a truly novel and pioneering character.

Parametric effects on Richtmyer–Meshkov instability of a V-shaped gaseous interface within linear stage

The Richtmyer–Meshkov instability of a V-shaped air/SF6 gaseous interface is numerically studied via a high-order finite difference scheme and a localized artificial diffusivity method. The oblique angle of the interface ranges from 20° to 75°, and the incident shock Mach number varies from 1.05 to 1.75. The wave patterns and the vortex structures are visualized during the interface evolution. A cavity is observed at the spike fingertip when the oblique angle decreases, which proves to be formed due to Mach reflection of the transmitted shock through velocity decomposition. By analyzing the linear growth rates of the interface, a modified empirical model for the reduction factor is suggested with model coefficients acquired by linear fitting for different Mach numbers. With shock polar analysis (SPA) method and visualization of the wave configuration, a criterion is proposed to explain the non-monotonic dependence of the linear growth rate on the oblique angle. In addition, Mach number effects on the linear growth rate are discussed by the SPA method, especially the anomalous behavior of the Mach 1.05 case.

A stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions for a 2D peridynamic diffusion model

In this work, a stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensional domain. To do so, we first use a high-order quadrature-based finite difference scheme to approximate the spatially nonlocal operator and use the BDF2 scheme to approximate the temporal direction. For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the discrete Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-type ABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green's first identity. For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete convolution sum form, then use the technique of the discrete orthogonal convolution kernels to obtain the optimal convergence order. Finally, numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

Conservative higher-order finite difference scheme for the coupled nonlinear Schrödinger equations

This paper introduces a conservative higher-order finite difference scheme for solving the coupled nonlinear Schrödinger equations. The Crank–Nicolson method is employed to discretize time derivatives and the sixth-order difference operator is used to discretize space derivatives, correspondingly, the resulting difference scheme has second-order accuracy in time and sixth-order accuracy in space. By utilizing the discrete energy method, the conservation of discrete mass and energy, the boundedness, existence and uniqueness of solution, unconditional stability and the convergence of the new scheme are proved. Then making use of Richardson extrapolation, the time accuracy is increased to the fourth order. Finally, the numerical experiments are conducted to validate the theoretical results presented in the paper.

A decoupled, linearly implicit and high-order structure-preserving scheme for Euler–Poincaré equations

It is challenging to develop high-order structure-preserving finite difference schemes for the modified two-component Euler–Poincaré equations due to the nonlinear terms and high-order derivative terms. To overcome the difficulties, we introduce a bi-variate function and carefully choose the intermediate average variable in the temporal discretization. Then, we obtain a decoupled and linearly implicit scheme. It is shown that the fully-discrete scheme can keep both the discrete mass and energy conserved. And the fully-discrete scheme has fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction. Several numerical examples are given to confirm the theoretical results.

Spatial-temporal high-order rotated-staggered-grid finite-difference scheme of elastic wave equations for TTI medium

In the finite-difference (FD) solution of the elastic wave equation for TTI medium, the traditional FD scheme can improve the computational accuracy of the spatial derivatives. However, the difference discretization on the temporal partial derivatives still uses the FD scheme of second-order accuracy, which cannot guarantee the computational accuracy of the temporal partial derivatives and often causes the temporal dispersion problem. The temporal high-order FD scheme can improve the computational accuracy of the temporal partial derivatives. Nevertheless, due to the complexity of the elastic wave equation in TTI medium, using the conventional staggered-grid technique to solve the elastic wave equation, some spatial derivatives need to be approximated by spatial interpolation, which will become new error and reduce the accuracy of the solution of the equations. Therefore, the currently commonly used temporal high-order FD method for the elastic wave equation in isotropic medium cannot be directly applied to the simulation of anisotropic medium. To solve the above problems, we propose a new temporal high-order rotated-staggered-grid FD stencil and derive a new temporal high-order FD scheme under this new stencil for the TTI medium elastic wave equation. Then we estimated the FD coefficients of the new stencil using Taylor series expansions (TE) and optimized the FD coefficients based on least squares (LS). Numerical experiments of the homogeneous TTI medium model and the modified BP model demonstrate that the TE + LS-based temporal high-order rotated-staggered-grid FD scheme can well solve the temporal dispersion in the numerical simulation of TTI medium.

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.

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

Higher-order asymptotic expansions and finite difference schemes for the fractional p-Laplacian

High fidelity simulation of dense vapours with thermodynamic consistent modelling

The paper describes the implementation of thermodynamic modelling within a high-order finite difference scheme. A new method to ensure thermodynamic consistency for Navier–Stokes simulations of real-gas11A real gas being one which does not obey the ideal gas law. flows is outlined. The paper shows that this method is suited to high fidelity simulations of dense-vapour flows with complex equations of state, such as found in Organic Rankine Cycle turbines. The method is illustrated with two examples of siloxane MM flow in a channel and in a turbine vane. The results show that thermodynamic consistency plays a vital role in the accurate prediction of irreversibility and therefore turbine loss.

A three level linearized compact difference scheme for a fourth-order reaction-diffusion equation

A high-order accuracy finite difference scheme is investigated to solve the one-dimensional extended Fisher-Kolmogorov (EFK) equation. A three level linearized compact finite difference scheme is proposed. Priori estimates and unique solvability are discussed in detail by the discrete energy method. The unconditional stability and convergence of the difference solution are proved. The new compact difference scheme has second-order accuracy in time and fourth-order accuracy in space in maximum norm. Numerical experiments demonstrate the accuracy, efficiency of our proposed technique.

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.