Second-order Central Difference Research Articles

Numerical study of the vortex-induced vibrations of a cantilevered pipe with a concentrated mass end using a wake oscillator model

In this paper, a numerical study based on a wake oscillator model has been carried out to determine the vortex-induced vibration (VIV) response characteristics of a flexible cantilevered pipe with a concentrated mass at the free end. First, a coupled model consisting of a cantilevered pipe and a wake oscillator is established, followed by discretization of the model and an iterative solution using the second-order central difference method. The displacement response and frequency response characteristics for different flow velocities and concentrated end masses have been compared. The numerical results reveal that the maximum value of the structural vibration displacement occurs at the bottom; i.e., free boundary end, for the cantilevered pipe with a concentrated mass at the free end. The dominant vibration mode typically shows a stepwise increase for an increase of the maximum flow velocity. As the vibration mode is transmitted from a lower order to a higher one, the vibration displacement of the structural end decreases fast. As the concentrated end mass increases, the dominant vibration mode shows a stepwise decrease, and as the vibration mode is transmitted from a higher order to a lower one, the vibration displacement of the structural end experiences a sharp increase.

A novel explicit fully-discrete momentum-preserving scheme of damped nonlinear stochastic wave equation influenced by multiplicative space-time noise

In this paper, the evolution laws of global momentum and averaged global momentum for the damped nonlinear stochastic wave equation (DNSWE) influenced by multiplicative space-time noise are derived. We innovatively combine the second-order central finite difference method and discrete gradient method in space, and integrate the splitting method and Störmer-Verlet type method in time. Both the novel spatial semi-discrete scheme and fully-discrete scheme constructed in this way can successfully preserve the corresponding evolution laws for discrete global momentum and discrete averaged global momentum. Numerical experiments on DNSWE with cubic nonlinearity validate the theoretical results.

Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results.

SeismicTransformer: An attention-based deep learning method for the simulation of seismic wavefields

Improving the accuracy and efficiency of seismic wavefield simulation aids geophysical problem-solving. The finite difference (FD) is widely used, but efficiency drops with increasing grids and higher order of difference formats. We propose an attention mechanism-based deep learning method called SeismicTransformer. Compared with theory-driven methods, such as the second-order central difference method, SeismicTransformer offers at least a tenfold improvement in speed. Compared with the networks without the attention mechanism, the SeismicTransformer achieves better results by utilizing global information. The proposed SeismicTransformer offers a promising solution for seismic wavefield simulation.

Numerical study of shock wave boundary layer interaction flows using a novel turbulence model

This study proposes a novel partial average fluctuation velocity (PAFV) turbulence based on the PAFV and the average strain rate tensor, which is used to calculate the shock wave boundary layer interaction flows, including cascade flow and transonic compressor flow. In order to illustrate the adequacy, the calculation results of several other turbulence models were also compared with PAFV, including k-ω, shear stress transport, SA-neg (negative Spalart-Allmaras), and stress-BSL (ω-based Reynolds stress model). The suitability of the PAFV turbulence model was initially tested by calculating the transonic flow in the L030-4 cascade. Subsequently, numerical computations were performed for the 3D (three-dimensional) National Aeronautics and Space Administration Rotor 67 transonic compressor rotor. In the numerical simulation, the flow control equations were transformed in a general curvilinear coordinate system for precision enhancement, and the spatial discretization of control equations was executed using the finite difference method. The convection term was processed using the Steger–Warming flux vector splitting method, the diffusion term was discretized using a second-order central difference scheme, and the time derivative term was discretized using the third-order Runge–Kutta method. A parallel computing strategy with multi-block partitioning was employed to speed up the calculations and facilitate faster convergence alongside a local time step method. All numerical computation procedures were written in Python, revealing the following: (1) The PAFV turbulence model aptly simulates the transonic flow within the cascade channel. The shock wave pattern in the cascade channel aligns well with the experimental schlieren images. (2) During the Rotor 67 transonic compressor rotor calculation, the flow–pressure ratio, flow efficiency, and inlet and outlet total temperatures and pressures align well with the experimental data. Furthermore, the resolution of the flow field structure related to the interaction between the tip leakage vortex and channel shock wave is high. (3) The turbulence model uses PAFV as the turbulence velocity scale, thereby suppressing the fluctuation velocity and turbulence viscosity near shock wave areas, preventing excessive turbulence viscosity. Moreover, PAFV inherently exhibits transport properties and anisotropic characteristics, making it well-suited for handling transonic compressor flows.

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Second-Order Central Difference Particle Filter Algorithm for State of Charge Estimation in Lithium-Ion Batteries

The estimation of the state of charge (SOC) in lithium-ion batteries is a crucial aspect of battery management systems, serving as a key indicator of the remaining available capacity. However, the inherent process and measurement noises created during battery operation pose significant challenges to the accuracy of SOC estimation. These noises can lead to inaccuracies and uncertainties in assessing the battery’s condition, potentially affecting its overall performance and lifespan. To address this problem, we propose a second-order central difference particle filter (SCDPF) method. This method leverages the latest observation data to enhance the accuracy and noise adaptability of SOC estimation. By employing an improved importance density function, we generate optimized particles that better represent the battery’s dynamic behavior. To validate the effectiveness of our proposed algorithm, we conducted comprehensive comparisons at both 25 °C and 0 °C under the new European driving cycle condition. The results demonstrate that the SCDPF algorithm exhibits a high accuracy and rapid convergence speed, with a maximum error which never exceeds 1.30%. Additionally, we compared the SOC estimations with both Gaussian and non-Gaussian noise to assess the robustness of our proposed algorithm. Overall, this study presents a novel approach to enhancing SOC estimation in lithium-ion batteries, addressing the challenges posed by the process itself and measurement noises.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

A second-order numerical method for nonlinear variable-order fractional diffusion equation with time delay

In this paper, a linearized numerical scheme of nonlinear variable-order fractional diffusion equation with time delay is constructed. We apply the L2−1σ formula to discretize the temporal derivative and second-order central difference scheme to discretize the spatial derivative. The proposed method is unconditionally stable and convergent with Oτ2+h2, where τ and h are the time and space steps, respectively. Numerical experiment demonstrates the effectiveness and accuracy of the numerical scheme.

Modelling Infectious Disease Dynamics: A Robust Computational Approach for Stochastic SIRS with Partial Immunity and an Incidence Rate

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings.

Numerical treatment of time-fractional sub-diffusion equation using p-fractional linear multistep methods

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are O ( τ p + h 2 ) for 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme.

Natural convection from solid and hollow cylinders with concave surface: A numerical approach

The buoyancy-driven thermofluid dynamics of a solid/hollow cylinder with concave surfaces have been studied in a laminar regime () using the finite-volume method to analyze the effects of ( and . Conservation equations for mass, momentum, and energy have also been solved in a 2-D axisymmetric geometry . Second-order upwind and central difference schemes have been deployed to discretize the convective and diffusive terms. Heat dissipation increases with irrespective of for both cylinders. However, the remains constant with the ratio for a hollow concave cylinder increases with whereas, decreases. Multiple vortices appear at the outside surface at a low (i.e. = 0.2), and these vortices almost disappear at high (i.e. = 0.6), affecting the outside average Nusselt number from low to high. In all cases, is higher than The thermofluid dynamics of solid and hollow cylinders have been delineated through different countour and vector plots, and two correlations for Nu have been deveoped.

Inverse asymptotic treatment: Capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities. These manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in (either single- or multi-phase) compressible flows. A plethora of bespoke discretization schemes have been developed to capture both types of discontinuities in physics-based simulations. However, when a quick response is required to rapidly emerging challenges such as the need to design novel hypersonic vehicles, the complexity of such implementations impedes a swift transition from problem formulation to computation. This is exacerbated by the need to compose multiple interacting physics, which may cause specialized schemes to break unexpectedly as governing equations need to be changed and/or added. We introduce “inverse asymptotic treatment” (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on standard, off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, rather than via reliance on specialized discretization schemes, IAT can seemlessly handle additional physics and thus makes it easier for novice end users to quickly obtain numerical results for a variety of multi-physics scenarios. This strategy also facilitates the use of a “multi-physics” compiler that automates the conversion of the modified PDEs to numerical source code readable by popular computational frameworks like OpenFOAM. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT’s ability to, within a span of just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA’s state-of-the-art VULCAN-CFD solver. Finally, we propose a novel LAD approach via “reverse-engineered” PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. Through canonical numerical tests, we demonstrate that this “limiter-inspired” LAD approach, when combined with second-order central differencing, can robustly and accurately model compressible flows.

Numerical Simulation and Characterization of Swirl Fluid Motion through Cylindrical Chambers

Axisymmetric solution of radial, axial and tangential components of velocity of flow through a cylindrical chamber with an inflow containing swirl and outflowis investigated numerically using Finite Volume Method. An in-house developed computer code for solving the three components of velocity and pressure is developed and validated using experimental data available from literature on the problem of axial vortex breakdown of confined flow in a cylinder due to rotation of one of the end walls. The code uses a staggered approach and fractional step projection method for decoupling the velocity and pressure. The convection and diffusion terms are approximated using a second-order accurate central difference scheme. This paper discusses the characteristics features of flow such as mixing streams, streams of fluid flowing close to the solid walls (cooling streams), swirl motion and development of flow structures.

Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model

In the last few years, reaction–diffusion models associated with discrete fractional calculus have risen in prominence in scientific fields, not just due to the requirement for numerical simulation but also due to the described biological phenomena. This work investigates a discrete equivalent of the fractional reaction–diffusion glycolysis model. The discrete fractional calculus tool is introduced to the discrete modeling of diffusion problems in the Caputo-like delta sense, and a fractional discretization diffusion model is described. The local stability of the equilibrium points in the proposed discrete system is examined. We additionally investigate the global stability of the equilibrium point by developing a Lyapunov function. Furthermore, this study indicates that the L1 finite difference scheme and the second-order central difference scheme can successfully preserve the characteristics of the associated continuous system. Finally, an equivalent summation representing the model’s numerical formula is shown. The diffusion concentration is further investigated for different fractional orders, and examples with simulations are presented to corroborate the theoretical findings.

Synchronization of Fractional Partial Difference Equations via Linear Methods

Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a fractional discrete nonlinear bacterial culture reaction-diffusion model using the Caputo h-difference operator and a second-order central difference scheme and an L1 finite difference scheme after deriving the discrete fractional version of the well-known Degn–Harrison system and Lengyel–Epstein system. Using appropriate techniques and the direct Lyapunov method, the conditions for full synchronization are determined.Furthermore, this research shows that the L1 finite difference scheme and the second-order central difference scheme may successfully retain the properties of the related continuous system. The conclusions are proven throughout the paper using two major biological models, and numerical simulations are carried out to demonstrate the practical use of the recommended technique.

NUMERICAL INVESTIGATION OF THE GROWTH- DIFFUSION MODEL

In this article, a numerical solution to the growth-diffusion problem is investigated by obtaining the results of computational experiments for the non-homogeneous growth-diffusion problem and finding its approximate solution by using the modified finite difference method. In this article, a numerical study is carried out by the modified finite difference method. The numerical scheme used a second-order central difference in space with a first-order in time.

Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays

This paper considers the dual hybrid effects of discrete-time stochastic genetic regulatory networks and discrete-space stochastic genetic regulatory networks in difference formats of exponential Euler difference and second-order central finite difference. The existence of a unique-weight pseudo-θ-almost periodic sequence solution for discrete-time and discrete-space stochastic genetic regulatory networks on the basis of discrete constant variation formulation is discussed, as well as the theory of semi-flow and metric dynamical systems. Furthermore, a finite-time guaranteed cost controller is constructed to reach global exponential stability of these discrete networks via establishing a framework of drive, response, and error networks. The results indicate that spatial diffusions of non-negative dense coefficients have no influence on the global existence of the unique weighted pseudo-θ-almost periodic sequence solution of the networks. The present study is a basic work in the consideration of discrete spatial diffusion in stochastic genetic regulatory networks and serves as a foundation for further study.

Numerical investigation of the entropy generation analysis for radiative MHD power-law fluid flow of blood through a curved artery with Hall effect

In the last few decades, mortality rate due to blood vessel diseases has hiked rapidly and it is almost one-third of total mortalities. Identifying the vascular geometry and hemodynamic blood flow inside the artery is essential for preventing vascular disease progression. By motivating vascular diseases effects on the human body, a mathematical model has been developed to discuss the blood flow phenomena and entropy estimation in the stenosed curved artery in the presence of a magnetic field and the Hall effect. The patient-specific condition of elliptic shape stenosis has been considered at the arterial wall. Blood is considered in the artery as two-phases; core and plasma region. Power-law fluid has been considered for core region fluid, while, Newtonian fluid is considered in the plasma region. The non-dimensional governing equations have been discretized using the second-order central difference method. Then, Stone's strongly implicit scheme is used in the ‘MATLAB’ software to solve a system of nonlinear partial differential equations (PDEs). A computing error tolerance of has been set for each iteration step throughout the computing process. The influence of various parameters such Hall parameter (m), Brinkman number (Br), flow behavior index (q), Radiation parameter (N), magnetic field (M), arterial curvature, etc., have been discussed graphically. The present study concludes that the heat transfer rate enhances with the increase in the Hall parameter values while wall shear stress reduces with it. In addition, the chances of stenosis deposition rise for highly curvature arteries, and also blood produces more entropy with the increase in viscous heating and thermal radiation intensity. Results from the current computational study will aid clinical researchers in making more accurate predictions about the prevalence of vascular diseases.

Numerical Study of Shock Wave Boundary Layer Interaction Based on an Anisotropic Turbulence Model

This paper presents numerical simulations of incident oblique shock wave impinging on a flat plate with a turbulent boundary layer using a nonlinear anisotropic turbulence model. Three cases with different incident shock angles under the same free-stream Mach number are studied, including attachment flow, incipient separation flow and outstanding separation flows. Convection terms and diffusion terms are calculated using the third-order Roe’s flux difference splitting scheme and the second-order central difference scheme in the code, respectively. The Runge-Kutta time marching method is employed to solve the governing equations for steady state solutions. Comparisons between the simulated results and experimental data are carried out, which include density contours, pressure distribution, skin friction coefficients distribution and streamline vectors. To capture the essential behavior of this complex anisotropic turbulence separation flow caused by the shock-wave-boundary-layer-interaction (SWBLI), an anisotropic eddy viscosity tensor and a nonlinear Reynolds stress-strain constitutive relationship are used in this model. The anisotropic eddy viscosity tensor is composed of turbulent fluctuation velocity vector and characteristic length vector which are obtained by statistical partial average scheme. With proper modeling of anisotropic eddy viscosity tensor, the numerical results are in good agreement with the experimental data. It is worthy to point it out that the computation accuracy decreases with increasing interaction intensity.