Central Finite Difference Scheme Research Articles

Numerical simulation of coupled Klein–Gordon–Schrödinger equations: RBF partition of unity

The coupled Klein–Gordon–Schrödinger equations have significant implications in quantum field theory, particle physics, cosmology, and nonlinear dynamics. In this study, we propose an efficient method for numerically simulating this system. The proposed approach involves employing the radial basis function partition of unity for spatial discretization. This method utilizes scaled Lagrange basis functions with polyharmonic spline kernels, taking advantage of the scalability property of the polyharmonic kernel to ensure stability in the approximation process. The resulting spatially discretized system yields a nonlinear time-dependent set of differential equations. To solve this system, we combine an implicit central finite difference scheme with a predictor–corrector procedure to overcome the nonlinearity. In the numerical results section, we assess the efficiency, accuracy, and versatility of the proposed method by conducting several simulations in large domains over extended periods. We present and compare these results with those obtained using alternative methods, demonstrating the effectiveness of the proposed approach in accurately solving the coupled Klein–Gordon–Schrödinger equations.

Evaluation of a high-order central-difference solver for highly compressible flows out of thermochemical equilibrium

A high-order shock-capturing central finite-difference scheme is evaluated for numerical simulations of hypersonic high-enthalpy flows out of thermochemical equilibrium. The scheme is an extension to thermochemical out-of-equilibrium flows of the technique presented in Sciacovelli et al. (2021) for high-speed flows in chemical nonequilibrium. It relies on a standard tenth-order accurate central-difference approximation of the inviscid fluxes, supplemented with a high-order accurate nonlinear artificial dissipation term of ninth-order accuracy in smooth flow regions. Close to flow discontinuities, a shock-capturing low-order term is activated based on a highly selective shock sensor. To enable robust and non oscillatory solutions in regions of strong discontinuities of the thermodynamic variables, including the vibrational temperature, a shock detector consisting of a combination of a pressure-based term and Ducros’ vorticity/dilatation sensor is applied to all conservation equations except that of vibrational energy. For the latter, a shock sensor based on the vibrational temperature itself is adopted instead, to account for the loose coupling between vibrational energy and pressure. The accuracy and robustness of the proposed approach is demonstrated for a variety of thermochemical non-equilibrium configurations, ranging from one-dimensional benchmarks to three-dimensional turbulent flows, for which the ILES capabilities of the selective high-order numerical dissipation are also showcased.

An extended quadratic auxiliary variable method for the singular Lennard-Jones droplet liquid film model

In this letter, we present an extended quadratic auxiliary variable method for a droplet liquid film model characterized by the singular Lennard-Jones potential. This innovative approach begins with the introduction of three new auxiliary variables, each satisfying a quadratic equation, with the primary objective of reformulating the original model into an equivalent system. The reformulated system exhibits remarkable properties, including the preservation of three quadratic invariants and a quadratic energy dissipation law, where these auxiliary variables satisfy consistent initial conditions. To solve it numerically, we employ the implicit midpoint method in time and the central finite difference scheme in space to produce a second-order fully discrete scheme for the original model. Under consistent initial conditions, our proposed scheme is rigorously proved to preserve the original energy dissipation law at the fully discrete level. Numerical experiments showcase the scheme’s accuracy and effectiveness. It is noteworthy that the quadratic auxiliary variable method is extended to gradient flow systems featuring rational energy functionals.

Hybrid conservative central/WENO finite difference scheme for two-dimensional detonation problems

In the present work, a hybrid finite difference scheme is proposed and tested for the study of two-dimensional detonation problems. The hybrid scheme consists of a fifth-order weighted essentially nonoscillatory (WENO) method for discontinuous regions and a fourth or sixth-order robust conservative central finite difference scheme for the remaining domain. An arbitrary shock sensor is used to identify the discontinuities. The proposed shock sensor is based on the absolute value of the density gradient with an arbitrary threshold computed at each time step using a fast image segmentation technique. The sensor is complemented with a Ducros sensor to avoid the selection of vortices as discontinuities. The hybrid scheme is tested for several benchmark examples, including a two-dimensional transverse detonation wave, detonation wave diffraction, and a multiple obstacles test case. The obtained results show that the proposed sensor detects discontinuities adequately and the hybrid schemes give the same results as the fifth-order WENO in faster computational time.

Natural Convection in a Newtonian Nanoliquid-Saturated Porous Enclosure with Local Thermal Non-Equilibrium Effect

Buoyancy-driven convective flow and heat transfer characteristics in a Newtonian nanoliquid-saturated porous square enclosure are analyzed numerically using a local thermal non-equilibrium model. An enclosure’s horizontal walls are considered free–free and adiabatic, and the vertical walls are free–free isothermal boundaries. The dimensionless governing equations are solved using a central finite difference scheme with second-degree accuracy, and the results are in satisfactory agreement with the earlier works. The impact of various parameters on streamlines and isotherms is analyzed and depicted graphically. The effect of Darcy number, thermal Rayleigh number, and the ratio of thermal conductivities slow down the liquid flow. The temperature distribution is maximum at sidewalls and diminishes the amount of heat transport. The opposite phenomenon is observed for the solute Rayleigh number and interphase transfer coefficient of liquid-particle phases. For large values of interphase heat transfer coefficients, liquid-solid and liquid-particle are said to be in the local thermal equilibrium phase. The amount of heat transfer increases with an increasing interphase heat transfer coefficient and the ratio of the phases’ thermal conductivities. Results of local thermal equilibrium situation can be obtained as the particular case of the study. The amount of heat transfer is maximum in the local thermal non-equilibrium situation, and enhanced by 0.09% compared with the local thermal equilibrium situation. Heat transport is 0.74% less in the sparsely packed porous medium compared with the low-porosity medium.

Applications of finite difference‐based physics‐informed neural networks to steady incompressible isothermal and thermal flows

AbstractPhysics‐informed neural networks (PINNs) have aroused an upsurge in scientific machine/deep learning community, especially in research areas involving solving partial differential equations. In this work, based on the multilayer perceptron neural network and finite difference (FD) method, we apply the finite difference‐based PINNs (FD‐PINNs) to simulate the steady incompressible flows. The spatial derivatives in nonlinear governing equations are discretized by the central FD scheme, and the physical boundary conditions can be exactly satisfied in a similar way to conventional fluid flow solvers. Compared to original PINNs using automatic differentiation (AD) to discretize the governing equations, FD‐PINNs hold three‐fold potential advantages in decoupling the predictive accuracy of derivatives from the network's approximating accuracy, fulfilling the physical boundary conditions, and saving computational costs when large‐size networks or meshes are used. We first attempt to make relatively fair comparisons between AD‐PINNs, FD‐PINNs and conventional flow solvers using a benchmark flow problem—steady incompressible laminar isothermal lid‐driven cavity flow across a wide range of Reynolds numbers (Re). Our results indicate that FD‐PINNs are superior to AD‐PINNs in both predictive precision, computational efficiency and training convergence. Moreover, FD‐PINNs exhibit higher accuracy than flow solvers when insufficient mesh points are used to solve high‐Re cavity flow. Even at Re = 10,000, with only 10 internal samples, FD‐PINNs can still precisely recover the flow details on a uniform mesh of size 51 × 51. Besides, the time consumption of FD‐PINNs grows much slower than that of conventional flow solvers. We also investigate the robustness of FD‐PINNs w.r.t. the noise level of samples and the effects of pressure boundary condition imposed on solid cavity walls. Furthermore, FD‐PINNs are extensively applied to predict the incompressible thermal flows: natural convective flows in a square cavity and in a concentric annulus, where the effects of governing equations on the predictive accuracy are discovered and discussed for the first time.

A numerical method for two-phase flow with its application to the Kelvin–Helmholtz instability problem

This article uses the artificial compressibility-based high-order finite difference method to simulate the two-phase Kelvin–Helmholtz (KH) instability problem. The equations are based on the mass-conserving Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. One of the advantages of the artificial compressibility approach is that many high-order numerical schemes based on hyperbolic conservation law can be applied. A fifth-order weighted essentially non-oscillatory (WENO) scheme is used for discretizing the convective terms while a standard central finite difference scheme is used for the viscous and surface tension terms. The system of equations is then solved using the Beam-Warming approximate factorization technique. For validation, the effects of both single- and double-mode sinusoidal perturbations on the Kelvin–Helmholtz instability dynamics are analyzed. When there is a single-mode sinusoidal perturbation, the interface roll-up at the center of the domain. Additionally, the role of the surface tension parameter in the instability’s dynamics is investigated. The development of Kelvin–Helmholtz instability is shown to be sensitive to the surface tension value. The analysis of grid convergence is also performed to capture the interface dynamics at varying resolutions. The comparison of the computed results is in good agreement with those from the literature. It is observed that the proposed technique effectively resolves the dynamics of the chaotically distorted interfaces of the Kelvin–Helmholtz instability in two-phase flow.

Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $$u\in C^{4}({\bar{\varOmega }})$$ is needed to preserve $${\mathcal {O}}(h^{2})$$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $${\mathcal {O}}(h^{\min (\sigma +\frac{1}{2}-\epsilon ,2)})$$ in both $$l^{2}$$ -norm and $$l^{\infty }$$ -norm in one-dimensional domain when the initial value and source term are both in $${\dot{H}}^{\sigma }(\varOmega )$$ but without any regularity assumption on the exact solution, where $$\sigma \ge 0$$ and $$\epsilon >0$$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $${\mathcal {O}}(h^{2})$$ in $$l^{2}$$ -norm and $${\mathcal {O}}(h^{\min (\sigma +\frac{3}{2}-\epsilon ,2)})$$ in $$l^{\infty }$$ -norm. It is worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and $${\mathcal {O}}(\tau ^{2})$$ convergence is obtained for all $$\alpha \in (0,1)$$ . Finally, some numerical experiments verify the effectiveness of the built theory.

Special Functions and Its Application in Solving Two Dimensional Hyperbolic Partial Differential Equation of Telegraph Type

In this article, we use the applications of special functions in the form of Chebyshev polynomials to find the approximate solution of hyperbolic partial differential equations (PDEs) arising in the mathematical modeling of transmission line subject to appropriate symmetric Dirichlet and Neumann boundary conditions. The special part of the model equation is discretized using a Chebyshev differentiation matrix, which is centro-asymmetric using the symmetric collocation points as grid points, while the time derivative is discretized using the standard central finite difference scheme. One of the disadvantages of the Chebyshev differentiation matrix is that the resultant matrix, which is obtained after replacing the special coordinates with the derivative of Chebyshev polynomials, is dense and, therefore, needs more computational time to evaluate the resultant algebraic equation. To overcome this difficulty, an algorithm consisting of fast Fourier transformation is used. The main advantage of this transformation is that it significantly reduces the computational cost needed for N collocation points. It is shown that the proposed scheme converges exponentially, provided the data are smooth in the given equations. A number of numerical experiments are performed for different time steps and compared with the analytical solution, which further validates the accuracy of our proposed scheme.

A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time . The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.

A Novel Finite Difference Scheme for Normal Mode Models in Underwater Acoustics

Normal mode models are commonly used to simulate sound propagation problems in horizontally stratified oceanic environments. Although several normal mode models have been developed, the fundamental techniques for accurately and efficiently solving the modal equation are still under development. Since the standard three-point central finite difference scheme (SFD) for the modal equation has a relatively large numerical error, at least twenty sampling grid points per wavelength should be set in the depth direction. Herein, a novel finite difference scheme (NFD) is developed to further improve the accuracy of the mode solution, and the resulting linear system still has a tridiagonal structure similar to that of the SFD. To compare the performance of the NFD to that of the SFD, the NFD has been implemented in the open-source normal mode model KrakenC, and three acoustic propagation cases have been carried out, including the plane-wave reflection, the Pekeris waveguide, and the Munk waveguide. Test results show that the NFD presented in this paper is more accurate than the SFD, and can be used to reduce the number of grid points needed in the depth direction for solving the modal equation in normal mode models.

Central random choice methods for hyperbolic conservation laws

AbstractIn this article, we briefly review the random choice method (RCM) and ADER methods for solving one and two-dimensional hyperbolic conservation laws. The main advantage of RCM is that it computes discontinuities with infinite resolution. In this method, the original problem is reduced to a set of local Riemann problems (RPs). The exact solutions of these RPs are used to form the solution of the original problem. However, RCM has the following disadvantages: (1) one should solve the RP exactly, however, the exact solutions are usually complex and unavailable for many problems. (2) The accuracy of the smooth region of the flow is poor. ADER methods are explicit, one-step schemes with a very high order of accuracy in time and space. They depend on the solution of the generalized RP (GRP) exactly. In Zahran (J Math Anal Appl 346:120–140, 2008), an improved version of ADER methods (central ADER) was introduced where the RPs were solved numerically and used central fluxes, instead of upwind fluxes. The improved central ADER schemes are more accurate, faster, simple to implement, RP solver free, and need less computer memory. To fade the drawbacks of the above schemes and keep their advantages, we propose, in this paper, an improved version of the RCM. We merge the central ADER technique with the RCM. The resulting scheme is called Central RCM (CRCM). The improvements are listed as follows: we use the WENO reconstruction for the initial data instead of constant reconstruction in RCM, we solve the RPs numerically by using central finite difference schemes and use random sampling to update the solution, as the original RCM. Here we use the staggered and non-staggered RCM. To enhance the accuracy of the new methods, we use a third-order TVD flux (Zahran in Bull Belg Math Soc Simon Stevin 14:259–275, 2007), instead of a first-order flux. Compared with the original RCM and the central ADER, the new methods combine the advantages of RCM, ADER, and central finite difference methods as follows: more accurate, very simple to implement, need less computer memory, and RP solver free. Moreover, the new methods capture the discontinuities with infinite resolution and improve the accuracy of the smooth parts. The new methods have less CPU time than the central ADER methods, this is due to less flux evaluation in CRCM. An extension of the schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented. We present several numerical examples for one and two-dimensional problems. The results confirm that the presented schemes are superior to the original RCM, ADER, and central ADER schemes.

Ion temperature profile importance in collisional sheath modelling

A plasma fluid model is being developed for the simulation of a direct current plasma discharge simulation including the sheath regions. The code uses a second order centered finite difference scheme and time integration is done by strong stability preserving third order Runge-Kutta method. The separation of scalar and vectorial quantities in two different grids gives stable results. After validation by comparison with theoretical ion sheath profiles, a one dimensional direct current argon discharge was simulated and compared to 1D3v particle-in-cell simulation results. It is shown that the inclusion of a non constant ion temperature profile is mandatory in fluid models in order to recover correct increase of ion velocity in sheaths and thus to simulate direct current (DC) discharges where collisions are not negligible in the sheaths.

Lateral-Torsional Instability and Biaxial Bending of Imperfect FRP I-Beams

This paper presents the outcome of a theoretical and experimental study of the behavior of Fiber Reinforced Polymer (FRP) I-beams exposed to lateral-torsional instability or when subjected to biaxial bending. Laboratory experiments involved the application of vertical and horizontal static loads to a 4 x 4 x ¼ in. I-beam with various lengths and the resulting deflections were recorded. Governing biaxial flexure and torsion differential equations were modified to account for the presence of initial imperfections and subsequently solved using a central finite-difference scheme. The theoretical predictions of the beam behavior were found to be in good agreement with what was observed in the laboratory.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.

A multi-splitting method to solve 2D parabolic reaction–diffusion singularly perturbed systems

In this paper we design and analyze a numerical method to solve a type of reaction–diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction–diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal.

Dynamic characteristics of elastic ring squeeze film damper coupled high-speed ball bearings

This paper explores an elastic ring squeeze film damper (ERSFD) model incorporating high-speed ball bearings to study the journal dynamics at different rotor operating speeds and squirrel cage stiffness. Firstly, dynamic model of the ERSFD is established. The journal works under ball-outer raceway excitation rather than rotor unbalance force. Bearing high-speed characteristics including ball centrifugal force, contact angle change and heavy thrust-applied load are outlined in quasi-static model when deriving bearing force. Oil film pressure is governed by the Reynolds equation and solved iteratively using the central finite difference scheme. Flow performance through the orifice and its effect on the oil film pressure are analyzed using computational fluid dynamics (CFD) technique. Elastic ring deformation is addressed using a thin-walled ring model. Secondly, squirrel cage stiffness and rotor operating speeds were chosen as influence parameters to numerically examine the journal steady-state and transient dynamics, and finally the center orbit and vibration of the journal were experimentally measured. The results demonstrate that an increase in squirrel cage stiffness enables the journal to orbit at small size but leads to higher transmissibility. Small squirrel cage stiffness and rotor operating speed result in high amplitude and velocity oscillations and destabilize the journal; alternatively, higher squirrel cage stiffness is ineffective in transmissibility attenuation regardless of rotor operating speed. It was concluded that a small squirrel cage stiffness cooperating with high rotor operating speed contributes to stabilize journal and attenuate vibration effectively.

High-order dimensionally-split Cartesian embedded boundary method for non-dissipative schemes

Centered finite-difference schemes are commonly used for high-fidelity turbulent flow simulations in canonical configurations because of their non-dissipative property and computational efficiency. However, their use in flow simulations over complex geometries is limited by the requirements of a structured grid and a stable boundary treatment in the absence of artificial (numerical) dissipation. Cartesian embedded boundary (EB) approaches provide an efficient structured-grid framework to apply difference schemes over complex domains. However, they are often restricted to low orders of accuracy because of numerical instabilities at the embedded boundaries and the issues of small-cell problem that are difficult to address with high-order accuracy. The present work discusses a systematic approach to obtain high-order EB methods with non-dissipative centered schemes in the interior. This approach, based on satisfying the primary and secondary conservation conditions, is employed to derive EB schemes that are up to sixth-order accurate in the interior and fourth-order accurate globally for hyperbolic, parabolic as well as incompletely parabolic problems. The proposed finite-difference discretization is, by construction, dimensionally split and addresses the small-cell problem without any cell/geometry transformations, thus, highly simplifying implementation in a flow solver. Various linear and non-linear numerical tests are performed to evaluate the stability and the accuracy of the proposed EB schemes.

Numerical analysis of a grating embedded bidirectional integrated optical coupler pressure sensor.

A numerical analysis of a grating embedded bidirectional optical coupled waveguide structure is presented for the first time, to our knowledge. A finite difference method (FDM) based scheme is devised to extract the allowed eigen TE and TM modes of the structure. Sensing characteristics of the grating employed between two high refractive index couplers are then explored. The influence of strain on the composite structure is numerically analyzed for better understanding of guiding phenomena. A numerical method based on a three-point central finite difference scheme with proper boundary conditions at the point of discontinuity is developed. For an accurate sensitivity analysis, a large number of mesh points (N=1000) are used in the FDM algorithm, while the whole analysis is done on MATLAB software. To the best of the authors' knowledge, Bragg grating sensitivities of individual TE and TM modes have been estimated for the first time. It is found that higher order TE and TM modes show improved sensitivity performance. The physics behind the improved sensitivity of the proposed structure is correlated with existing cases. The proposed technique is based on effective refractive index theory, and hence it is easy to implement. This work can be easily extended to obtain temperature, humidity, and vibration sensitivities of other novel structures.

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.