Fourth-order Spatial Accuracy Research Articles

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

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.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

A linearlized mass-conservative fourth-order block-centered finite difference method for the semilinear Sobolev equation with variable coefficients

In this paper, a two-dimensional variable-coefficients semilinear Sobolev equation under Neumann boundary condition is considered, and a novel Crank–Nicolson type linearized fourth-order block-centered finite difference scheme which preserves mass is developed and analyzed. Under local Lipschitz continuous assumption on the nonlinear source term, second-order temporal and fourth-order spatial accuracy are rigorously proved for the primal scalar variable p, its gradient u and its flux q simultaneously. Then, stability under a rough time stepsize condition is proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions and well performance of the proposed scheme.

Numerical Algorithm with Fourth-Order Spatial Accuracy for Solving the TimeFractional Dual-Phase-Lagging Nanoscale Heat Conduction Equation

Numerical Algorithm with Fourth-Order Spatial Accuracy for Solving the TimeFractional Dual-Phase-Lagging Nanoscale Heat Conduction Equation

Eilmer: An open-source multi-physics hypersonic flow solver

This paper introduces Eilmer, a general-purpose open-source compressible flow solver developed at the University of Queensland, designed to support research calculations in hypersonics and high-speed aerothermodynamics. Eilmer has a broad userbase in several university research groups and a wide range of capabilities, which are documented on the project's website, in the accompanying reference manuals, and in an extensive catalogue of example simulations. The first part of this paper describes the formulation of the code: the equations, physical models, and numerical methods that are used in a basic fluid dynamics simulation, as well as a handful of optional multi-physics models that are commonly added on to do calculations of hypersonic flow. The second section describes the processes used to develop and maintain the code, documenting our adherence to good programming practice and endorsing certain techniques that seem to be particularly helpful for scientific codes. The final section describes a half-dozen example simulations that span the range of Eilmer's capabilities, each consisting of some sample results and a short explanation of the problem being solved, which together will hopefully assist new users in beginning to use Eilmer in their own research projects. Program summaryProgram Title: EilmerCPC Library link to program files:https://doi.org/10.17632/gy2ds2fyxm.1Developer's repository link:https://github.com/gdtk-uq/gdtkCode Ocean capsule:https://codeocean.com/capsule/7226427Licensing provisions: GPLv3Programming language: D, LuaSupplementary material:https://gdtk.uqcloud.netNature of problem: Eilmer solves the compressible Navier-Stokes equations with a particular emphasis on flows at hypersonic speeds. The code includes modelling for high-temperature gas effects such as chemical and vibrational nonequilibrium. Eilmer can be used for the simulation for unsteady and steady flows.Solution method: The code is implemented in D [1] and built on a finite-volume formulation that is capable of solving the Navier-Stokes equations in 2D and 3D computational domains, discretised with structured or unstructured grids. Grids may be generated using a built-in parametric scripting tool or imported from commercial gridding software. The inviscid fluxes are computed using the reconstruction-evolution approach. In structured-grid mode, reconstruction stencils up to fourth-order spatial accuracy are available. In unstructured-grid mode, least-squares reconstruction provides second-order spatial accuracy. A variety of flux calculators are available in the code. Viscous fluxes are computed with compact stencils with second-order spatial accuracy. For unsteady flows, explicit time-stepping with low-order RK-family schemes are available, along with a point-implicit Backward-Euler update scheme for stiff systems of equations. For steady flows, convergence can be greatly accelerated using a Jacobian-free Newton-Krylov update scheme, which seeks a global minimum in the residuals using a series of large pseudo-timesteps. Domain decomposition is used for parallel execution using both shared memory and distributed memory programming techniques.Additional comments including restrictions and unusual features: Eilmer provides a programmable interface for pre-processing, post-processing and user run-time customisations. The programmable interface is enabled using a built-in embedded interpreter for the Lua programming language [2]. Run-time customisations include used-defined boundary conditions, source terms and grid motion.

Numerical Method to Modify the Fractional-Order Diffusion Equation

Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: 1 + 1 / 12 δ x 2 p j k − 1 + 1 / 12 δ x 2 p j k − 1 = μ α ∑ l = 0 k λ α l δ x 2 p j k − l + μ β ∑ l = 0 k λ β l δ x 2 p j k − l + τ / 2 1 + 1 / 12 δ x 2 f j k − 1 + f j k , k = 1,2 , … , K ; j = 1,2 , … , j − 1 , p j 0 = ω j , j = 0,1 , … , J , p 0 k = φ t k , p j k = ψ t k , k = 0,1 , … , K , f j l = f x j , t l , ω j = ω x j . Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in ∂ p x , t / ∂ t = ∂ 1 − α / ∂ t 1 − α + B ∂ 1 − β / ∂ t 1 − β ∂ 2 p x , t / ∂ x 2 + f x , t , 0 < t ≤ 1,0 < x < 1 , p x , 0 = 0,0 ≤ x ≤ 1 . p 0 , t = t 2 , p 1 , t = e t 2 , 0 ≤ t ≤ 1 . Our suggested method can improve the time precision with a certain value.

High order accurate in time, fourth order finite difference schemes for the harmonic mapping flow

In this paper, a fully discrete numerical scheme is proposed and analyzed for the harmonic mapping flow, with the fourth order spatial accuracy and higher than third order temporal accuracy. The fourth order spatial accuracy is realized via a long stencil finite difference, and the boundary extrapolation is implemented by making use of higher order Taylor expansion. Meanwhile, the high order (third or fourth order) temporal accuracy is based on a semi-implicit algorithm, which uses a combination of explicit Adams–Bashforth extrapolation for the nonlinear terms and implicit Adams–Moulton interpolation for the viscous diffusion term, with the corresponding integration formula coefficients. Both the consistency, linearized stability analysis and optimal rate convergence estimate (in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm) are provided. A few numerical examples are also presented in this article.

A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

Enhanced solution of 2D incompressible Navier–Stokes equations based on an immersed-boundary generalized harmonic polynomial cell method

Poisson equation lies in the heart of the numerical solutions of incompressible Navier–Stokes equations (NSEs) based on the popular projection methods, which decouple the pressure and velocities fields. This paper presents enhanced numerical solutions of the 2D incompressible NSEs inspired by a newly-developed Generalized Harmonic Polynomial Cell (GHPC) method for the Poisson equation, which has a fourth-order spatial accuracy. To achieve that, the original GHPC method is adapted to accommodate immersed boundaries, necessary when a staggered grid for velocities and pressure is applied. The finite difference method (FDM) is used in the spatial discretization of the diffusion and advection terms. Numerical analyses of lid-driven cavity flow, a Taylor–Green vortex, and flow around a smooth circular cylinder all show encouraging results, confirming the accuracy and efficiency of the new solver. Through comparison with an NSEs solver which applies second-order FDM for the Poisson equation, we demonstrate that the accuracy of the pressure solution can be greatly improved by applying the more accurate GHPC method for the Poisson equation, even though the accuracy of the velocities solutions are limited by the numerical schemes for advection–diffusion equations. The accuracy in the pressure solution can also be translated into CPU time saving to achieve a predefined accuracy. The present immersed-boundary GHPC Poisson equation solver can easily be ‘plugged’ into other existing NSEs solvers utilizing staggered grids.

Maximum norm error estimates of a linear CADI method for the Klein–Gordon–Schrödinger equations

In this paper, an efficient linear compact alternating direction implicit (CADI) scheme with second order temporal accuracy and fourth order spatial accuracy is proposed for approximating solution to the initial–boundary problem for the two-dimensional Klein–Gordon–Schrödinger equations. Furthermore, maximum norm error estimates of numerical solutions are obtained by using the energy method and the mathematical induction method. Finally, some numerical experiments are presented to support the theoretical results.

Discrete-time orthogonal spline collocation method for a modified anomalous diffusion equation

The modified anomalous sub-diffusion equation (MASDE) describes processes that become less anomalous as time progresses. In this paper, based on Crank-Nicolson and weighted and shifted Grünwald difference (WSGD) formula, we develop orthogonal spline collocation (OSC) method with second-order temporal accuracy and fourth-order spatial accuracy for MASDE. The stability and convergence of our numerical schemes are analysed in detail. Also, our numerical results are consistent with our theoretical analysis.

Novel numerical method of the fractional cable equation

In this article, mainly based on the second order compact approximation of first order derivative, the novel numerical method with second order temporal accuracy and fourth order spatial accuracy is proposed to solve the fractional cable equation. The numerical analysis involving convergence and stability of the novel numerical method subject to strict and detailed discussion. In addition, the numerical experiment strongly support the theoretical analysis results.

Prediction of evanescent coupling efficiency in two parallel silica nanowires

In this study the efficiency of evanescent wave coupling between two air-clad silica nanowires in single-mode operation is numerically predicted in time-domain using the finite difference method. To this end, a three-dimensional scheme developed for solving the Maxwell’s equations on staggered grids will be employed. The electric and magnetic field solutions are sought subjected to the zero-divergence condition (or Gauss’s law) in discrete level. In addition, we aim to conserve Hamiltonians existing in ideal Maxwell’s equations at all times by adopting the explicit second-order accurate symplectic partitioned Runge–Kutta temporal scheme. Moreover, all the spatial derivative terms in the Faraday’s and Ampère’s equations are approximated by the proposed scheme which can render not only fourth-order spatial accuracy, but also minimize the discrepancy between the exact and the derived numerical phase velocities.

High-Order Finite-Volume Method with Block-Based AMR for Magnetohydrodynamics Flows

A high-order central essentially non-oscillatory (CENO) finite volume scheme combined with a block-based adaptive mesh refinement (AMR) algorithm is proposed for the solution of the ideal magnetohydrodynamics equations. The high-order CENO finite-volume scheme is implemented with fourth-order spatial accuracy within a flexible multi-block, body-fitted, hexahedral grid framework. An important feature of the high-order adaptive approach is that it allows for anisotropic refinement, which can lead to large computational savings when anisotropic flow features such as isolated propagating fronts and/or waves, shocks, shear surfaces, and current sheets are present in the flow. This approach is designed to handle complex multi-block grid configurations, including cubed-sphere grids, where some grid blocks may have degenerate edges or corners characterized by missing neighboring blocks. A procedure for building valid high-order reconstruction stencils, even at these degenerate block edges and corners, is proposed, taking into account anisotropic resolution changes in a systematic and general way. Furthermore, a non-uniform or heterogeneous block structure is used where the ghost cells of a block containing the solution content of neighboring blocks are stored directly at the resolution of the neighbors. A generalized Lagrange multiplier divergence correction technique is applied to achieve numerically divergence-free magnetic fields while preserving high-order accuracy on the anisotropic AMR grids. Parallel implementation and local grid adaptivity are achieved by using a hierarchical block-based domain partitioning strategy in which the connectivity and refinement history of grid blocks are tracked using a flexible binary tree data structure. Physics-based refinement criteria as well as the CENO smoothness indicator are both used for directing the mesh refinement. Numerical results, including solution-driven anisotropic refinement of cubed-sphere grids, are presented to demonstrate the accuracy and efficiency of the approach.

A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel

ABSTRACTIn this paper, a fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel is developed and analysed. Compact difference scheme is used as a high order approximation for spatial derivative in the fractional parabolic equation, and the Caputo–Fabrizio (C–F) fractional derivative is discretized by a second-order approximation. We have proved that the proposed scheme has fourth-order spatial accuracy and second-order temporal accuracy.However, due to the nonlocal nature of fractional operator, numerically solving the time fractional parabolic equation with traditional direct solvers generally require memories and computational complexity, where N and M represent the number of time steps and grid points in space, respectively. We developed a fast evaluation scheme for the new C–F fractional derivative, which significantly reduced the computational complexity to , and the memory requirement to . Numerical experiments are given to verify the effectiveness and high order convergence of the proposed scheme.

High-order detached eddy simulation of unsteady flow around NREL S826 airfoil

Delayed detached eddy simulation (DDES) has been a common approach for unsteady turbulent flow problems, but it has several weaknesses which limit its use to detached flows. On the other hand, some enhancements introduced recently have made it possible to apply to attached flows as well. In the present investigation, flow with relatively low Reynolds numbers of 100,000 and 145,000 around NREL S826 airfoil is simulated using DDES models with two different length scale definitions: standard length scale and shear-layer-adapted length scale (SLADDES). The solver, originally developed for aeroacoustic simulations, features fourth-order spatial accuracy enhanced with symmetry-preserving and dispersion-relation-preserving characteristics. Simulations are done at various angles of attack around stall regions. Results are compared with other simulations and measurements. The present results indicate that SLADDES accelerates transition from modeled turbulence mode to resolved turbulence mode even in attached flows. Consequently, SLADDES shows aerodynamic results in better agreement with the experiments than the standard DDES, and allows coarser grids thanks to low sensitivity to spanwise grid spacings. Moreover, it appears that both DDES and SLADDES approaches benefit from the low-dissipation, low-dispersion, and high-order scheme of the solver.

Lattice Boltzmann Simulation of Transient Natural Convection in a Staggered Cavity with Four Vertically Heated Walls

The work in this manuscript deals with the numerical simulation of natural convection in a staggered cavity with the help of a recently developed two-dimensional double Multiple-Relaxation-Time (MRT) thermal Lattice Boltzmann method (LBM). In the last decade, there has been a rapid rise in the development of Lattice Boltzmann methods. However, its application in the simulation of natural convection from a staggered cavity has been carried out for the first time in this study. A careful undermining into the existing literature of heat and mass transfer reveal that study of natural convections in cross-sectional cavities is notably absent. Therefore, in this manuscript, we attempt to review the recently developed method and tried to analyze its implementation on natural convection in a staggered cavity with four differentially heated vertical walls. The problem geometry has eight boundaries. It is a staggered cavity with adiabatic horizontal walls and differentially heated vertical walls. The flow inside the thermally driven staggered cavity has been carefully studied for Rayleigh numbers 103, 104 and 105. The velocity and pressure boundary conditions are determined by a non-equilibrium extrapolation rule. As no benchmark results are available in the literature for this relatively new problem, we carry out its simulation with the help of a yet another well established scheme. This scheme is a higher-order compact (HOC) scheme with fourth order spatial accuracy and second order temporal accuracy. Our results show that there is a very good agreement between both these methods which exemplifies the accuracy and credibility of our results.

A compact ADI method and its extrapolation for time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions

A compact alternating direction implicit (ADI) finite difference method is proposed for two-dimensional time fractional sub-diffusion equations with nonhomogeneous Neumann boundary conditions. The unconditional stability and convergence of the method is proved. The error estimates in the weighted L2- and L∞-norms are obtained. The proposed method has the fourth-order spatial accuracy and the temporal accuracy of order min{2−α,1+α}, where α∈(0,1) is the order of the fractional derivative. In order to further improve the temporal accuracy, two Richardson extrapolation algorithms are presented. Numerical results demonstrate the accuracy of the compact ADI method and the high efficiency of the extrapolation algorithms.

Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation

In this paper, based on the second order compact approximation formula of first order derivative, the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy is developed to simulate the modified fractional diffusion equation; the convergence, stability and solvability of the numerical simulation method are analyzed by Fourier analysis and algebraic theory, respectively; the theoretical results extremely consistent with the numerical experiment.