Standard Central Difference Research Articles

PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

Evaluating the Electromagnetic Field Problem Generated by the Power Transmission Line Using High Order Compact Method

This paper compares a finite difference method-based computational scheme for evaluating electromagnetic field problems in power transmission lines in cases of on surface and in Ground. The scheme uses Maxwell's partial differential equations to represent electric and magnetic field components and approximate boundary conditions. The High Order Compact Method (HOC) is applied to estimate the electric field (Ez )in ground and compares it with the standard central difference scheme. The HOC produces more accurate results than the traditional central approximation at 99.7% for the electric field Ez. It also calculates both of Electric and magnetic intensity to evaluate the effect of Electric and magnetic intensities in ground and on surface respect to distance.

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.

CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

A dispersion analysis of uniformly high order, interior and boundaries, mimetic finite difference solutions of wave propagation problems

A preliminary stability and dispersion study for wave propagation problems is developed for mimetic finite difference discretizations. The discretization framework corresponds to the fourth-order staggered-grid Castillo-Grone operators that offer a sextuple of free parameters. The parameter-dependent mimetic stencils allow problem discretization at domain boundaries and at the neighbor grid cells. For arbitrary parameter sets, these boundary and near-boundary mimetic stencils are lateral, and we here draw first steps on the parametric dependency of the stability and dispersion properties of such discretizations. As a reference, our analyses also present results based on Castillo-Grone parameters leading to mimetic operators of minimum bandwidth that have been previously applied in similar physical problems. The most interior parameter-dependent mimetic stencils exhibit a specific Toeplitz-like structure, which reduces to the standard central finite difference formula for staggered differentiation at grid interior. Thus, our results apply to the whole discretization grid. The study done for the 1-D problem could be applied to the discretization of a free surface boundary condition along an orthogonal gridline to this boundary.

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.

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

High-Order WENO-based Semi-Implicit Projection Method for Incompressible Turbulent Flows: Development, Accuracy, and Reynolds Number Effects

A novel high-order projection method is presented in this study for the simulation of incompressible turbulent flows, utilising weighted essentially non-oscillatory (WENO) schemes. Spatial discretization involves using WENO schemes for nonlinear convection and standard central differences (CD) for the viscous term. The approach combines various orders of WENO and central difference schemes (WENO3/CD2, WENO5/CD4, and WENO7/CD6) to achieve third, fifth, and seventh-order spatial accuracy for velocity fields, respectively. Validating this method involved a thorough examination using the 2D Taylor Green vortex problem in both space and time, along with simulations of 3D Taylor-Green vortex problems at increasing Reynolds numbers (Re) of 1, 600, 16, 000, and 160, 000. The outcomes encompassed analyses of kinetic energy, enstrophy, energy spectra, and vortex fields, indicating the method's effectiveness in accurately simulating high Reynolds number turbulent flows without additional sub-grid scale models.

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 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.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

The finite difference method for the fourth‐order partial integro‐differential equations with the multi‐term weakly singular kernel

This paper is concerned with an efficient numerical method for a class of fourth‐order partial integro‐differential equations (PIDEs) with weakly singular kernel. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time . The proposed method is constructed on the graded meshes, which can achieve the second‐order convergence in time for weakly singular solutions. The product integral rule of the Riemann–Liouville fractional integral with the generalized time‐stepping is used for the time discretization, and the standard central difference formula is used for the space discretization. The stability and convergence of the method are proved, and the optimal error estimates in the discrete ‐norm and ‐norm are obtained. Numerical results show the effectiveness of the proposed method. Further, by the extrapolation method, we improve the convergence order in space and time to four order, respectively.

Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank–Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann–Liouville fractional integral term. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at t=0 so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, the stability and convergence are proved by means of the energy argument. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.

An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method.

Generalized SAV-Exponential Integrator Schemes for Allen--Cahn Type Gradient Flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen-Cahn equation. While some commonly-used first-order time stepping schemes have turned out to preserve unconditionally both energy dissipation law and MBP for the equation, restrictions on the time step size are still needed for existing second-order or even higher-order schemes in order to have such simultaneous preservation. In this paper, we develop and analyze novel first- and second-order linear numerical schemes for a class of Allen-Cahn type gradient flows. Our schemes combine the generalized scalar auxiliary variable (SAV) approach and the exponential time integrator with a stabilization term, while the standard central difference stencil is used for discretization of the spatial differential operator. We not only prove their unconditional preservation of the energy dissipation law and the MBP in the discrete setting, but also derive their optimal temporal error estimates under fixed spatial mesh. Numerical experiments are also carried out to demonstrate the properties and performance of the proposed schemes.

Numerical prediction of vortex-induced vibrations of a long flexible riser with an axially varying tension based on a wake oscillator model

A numerical study based on the wake oscillator model has been conducted to determine the vortex-induced vibration (VIV) responses of a flexible riser with an axial time-varying tension. Three different types of flows, viz., linear shear, exponential shear, and real stepped flows, have been considered. The coupling equations of a structural oscillator and a wake oscillator have been solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, displacement envelope, and displacement evolution for three different flow profiles have been systematically compared. Subsequently, for each flow profile, the effect of the different tension models on the VIV characteristics has been investigated. The numerical results indicated that, both the VIV displacement and VIV frequency in real stepped flow had diverse characteristics from those in linear shear flow. Compared with the corresponding VIV characteristics in real stepped flow, the VIV displacement in exponential shear flow was similar, however, the VIV frequency in exponential shear flow was disparate. The VIV displacement with the constant tension model had larger values than that with the real tension model.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

An implicit difference scheme for the fourth-order nonlinear partial integro-differential equations

In this paper, an implicit difference scheme is constructed and analyzed for the fourth-order nonlinear partial integro-differential equations. The backward Euler scheme and convolution quadrature are combined for time derivative and Riemann–Liouville (R–L) fractional integral terms. A fully discrete difference scheme is established with the space discretization by the standard central difference approximation. For nonlinear convection term , by the Galerkin method based on piecewise linear test functions, we can handle nonlinear term implicitly and attain a system of nonlinear algebraic equations. The stability and convergence are derived by the energy method. Especially, the existence and uniqueness of the numerical solutions are strictly proved for the nonlinear system. Also, we introduce three methods for solving the nonlinear systems, where two existing methods and a new linearized iterative algorithm are given to show the efficiency of our scheme. Numerical results are consistent with the theoretical analysis.

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>