Third-order Difference Research Articles

Simulation of advection\u2013diffusion\u2013dispersion equations based on a composite time discretization scheme

In this work, we develop a high-order composite time discretization scheme based on classical collocation and integral deferred correction methods in a backward semi-Lagrangian framework (BSL) to simulate nonlinear advection–diffusion–dispersion problems. The third-order backward differentiation formula and fourth-order finite difference schemes are used in temporal and spatial discretizations, respectively. Additionally, to evaluate function values at non-grid points in BSL, the constrained interpolation profile method is used. Several numerical experiments demonstrate the efficiency of the proposed techniques in terms of accuracy and computation costs, compare with existing departure traceback schemes.

Universal odd-even staggering in isotopic fragmentation and spallation cross sections

In a recent work [Phys. Rev. C 97, 044619 (2018)], a universal odd-even staggering (OES) has been observed in isotopic cross sections of neutron-rich nuclei produced by various fragmentation and spallation reactions. However, the universality of this OES is not clear for neutron-deficient nuclei, and thus more quantitative studies of this OES effect are required for these nuclei from different reactions. For neutron-deficient nuclei with ($N\ensuremath{-}Z$) from 0 to $\ensuremath{-}4$, the OES magnitudes are calculated by a third-order difference formula using cross sections measured in various fragmentation and spallation systems and they are applied to derive the evaluation values of the OES magnitudes. The evaluated OES magnitudes for both neutron- and proton-rich nuclei are benchmarked with some additional experimental data from fragmentation as well as spallation reactions. Furthermore, the OES factors predicted by two recent fragmentation and spallation models are checked by comparing with the OES magnitudes evaluated from extensive experimental data. This comparison suggests that the OES factors in these fragmentation and spallation models should be improved to reproduce measured cross sections.

Improvement of third-order finite difference WENO scheme at critical points

ABSTRACTIn the paper, an improved third-order finite difference weighted essentially non-oscillatory scheme is presented for achieving the optimal order near critical points. A new global smoothness indicator is obtained with the way of Taylor expansion for the local smoothness indicator. In the framework of the conventional WENO-Z scheme, we present an improved third-order finite difference WENO scheme. The proposed scheme is verified to achieve third-order accuracy by several benchmarks and the behaviour of the present scheme is proved on a variety of one- and two- dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with the other third-order WENO schemes.

Cabaret Difference Scheme with Improved Dispersion Properties

A difference scheme for the transfer problem, constructed as a linear combination of the Cabaret scheme and the central difference scheme, is proposed. The stability and dispersion properties of the scheme are studied. It is shown that the constructed scheme has the best dispersion properties for high-frequency harmonics at small Courant numbers compared with the known Cabaret scheme for the transport equation. A comparison is made of the errors of this scheme and the two-parameter third-order difference accuracy scheme based on the numerical experiments on the previously used test problem sets. It is shown that in the normal grid space L1 the developed scheme has smaller errors and also uses a more compact template (in calculating the inode the node values i – 1, i, and i + 1 are used), and the transition to the next time layer is carried out for a smaller number of arithmetic operations.

Oscillatory Behavior of Third-Order Nonlinear Difference Equations with a Nonlinear-Nonpositive Neutral Term

We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form: $$\begin{aligned} \Delta \left( \left( a(t)\left( \Delta ^{2}\left( x(t)-p(t)x^{\alpha }(t-k) \right) \right) ^{\gamma } \right) +q(t)x^{\beta }(t-m+1)=0,\right. \end{aligned}$$ with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with $$\hbox {p (t)} = 0$$ . Examples are given to illustrate the main results.

Boundedness properties of the generalized Todd's difference equation

ABSTRACTWe consider the third-order difference equation with and We show that if then there exist positive initial conditions such that the solutions are unbounded except for the case and

Numerical simulation of power transmission lines

A complete mathematical model of power transmission lines is obtained for time and space dependent shunt parameters and time dependent series parameters. All possible initial and boundary conditions related with line terminations are tabulated. The nonlocal boundary value problem for the resulting telegraph equation is considered and numerical solutions of this problem are computed by using various finite difference schemes. Errors in the numerical solutions and the relevant central processing unit (CPU) time spent for processing are presented. The problem is also solved by using perturbation-iteration algorithm (PIA). The results are compared with each others and the advantage or disadvantage of any method is discussed. Second-order difference method and third-order difference method are applied for solving one-dimensional telegraph equation used for power transmission lines in the case of open-circuit ended and short-circuit ended. The results about the voltage responses of all cases are discussed.

The Third-Order Approximation Godunov Method for the Equations of Gas Dynamics

The paper suggests a new modification of Godunov difference method with the 3rd order approximation in space and time for hyperbolic systems of conservation laws. The diёerence scheme uses the simultaneous discretization of the equations in space and time without of Runge — Kutta stages. An exact or approximate solution of Riemann problem is applied to calculate numerical fluxes between cells. Before the time step, corrections to the arguments of the Riemann problem providing third-order approximations for linear systems are calculated. After the time step, the numerical solution correction procedure is applied to eliminate the second-order error arising from the nonlinearity of the equations. The paper presents the results of experimental numerical verification of the method approximation order on the exact smooth solution inside the fan of the expansion wave. The test results completely confirm the third order of the presented method. The proposed approach of constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.

Boundedness and Asymptotic Behaviour of the Solutions for a Third- Order Fuzzy Difference Equation

Our aim in this paper is to investigate the dynamics of a third-order fuzzy difference equation. By using new iteration method for the more general nonlinear difference equations and inequality skills as well as a comparison theorem for the fuzzy difference equation, some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for the nonlinear fuzzy system are obtained. Moreover, some numerical solutions of the equation describing the system are given to verify our theoretical results.

On the boundedness of solutions of a class of third-order rational difference equations

ABSTRACTWe consider the difference equation with and If we show that for all positive initial conditions the solutions of the difference equations are bounded. If then there exists positive initial conditions such that the solutions are unbounded.

Lie symmetry analysis and explicit formulas for solutions of some third-order difference equations

Lie group theory is applied to rational difference equations of the formwhere (an)n∈ℕ0, (bn)n∈ℕ0 are non-zero real sequences. Consequently, new symmetries are derived and exact solutions, in unified manner, are constructed. Based on some constraints in the expression of the symmetry generators, we split these solutions into different categories. This work generalises a recent result by Ibrahim [9].

Universal odd-even staggering in isotopic fragmentation and spallation cross sections of neutron-rich fragments

An evident odd-even staggering (OES) in fragment cross sections has been experimentally observed in many fragmentation and spallation reactions. However, quantitative comparisons of this OES effect in different reaction systems are still scarce for neutron-rich nuclei near the neutron drip line. By employing a third-order difference formula, the magnitudes of this OES in extensive experimental cross sections are systematically investigated for many neutron-rich nuclei with ($N\ensuremath{-}Z$) from 1 to 23 over a broad range of atomic numbers ($Z\ensuremath{\approx}3--50$). A comparison of these magnitude values extracted from fragment cross sections measured in different fragmentation and spallation reactions with a large variety of projectile-target combinations over a wide energy range reveals that the OES magnitude is almost independent of the projectile-target combinations and the projectile energy. The weighted average of these OES magnitudes derived from cross sections accurately measured in different reaction systems is adopted as the evaluation value of the OES magnitude. These evaluated OES magnitudes are recommended to be used in fragmentation and spallation models to improve their predictions for fragment cross sections.

ON A THIRD ORDER DIFFERENCE EQUATION

ON A THIRD ORDER DIFFERENCE EQUATION

Symmetry analysis and conservation laws of some third-order difference equations

We adopt a procedure for determining the continuous (Lie) symmetries for third-order difference equations and utilize these to reduce the order of the equations – this reduction leads to some solutions of the equations under investigation. We further investigate the existence of first integrals of the third order equations using a, now, established, procedure.

Dynamical Behavior of a Third-Order Difference Equation with Arbitrary Powers

Dynamical Behavior of a Third-Order Difference Equation with Arbitrary Powers

The Effect of Aliased and De-aliased Formulation for DNS Analysis in Plane Poiseuille Flow

In this research, the performance of various aliased and de-aliased schemes was studied in pseudo-spectral (PS) simulation. The PS method was used to perform aliased and de-aliased direct numerical simulation of a turbulent plane Poiseuille channel flow. The effects of aliasing errors in Navier–Stokes equations were examined for both common skew-symmetric and rotational forms. Due to boundary conditions, Fourier series in periodic directions and Chebyshev expansion in normal direction were considered. A third-order backward difference scheme was used for temporal approximation method, and different types of “2/3 rule” de-aliasing procedures in various directions were examined. The results for aliased and de-aliased formulation showed that generally, the skew-symmetric form gives the most accurate results. But de-aliased rotational form was more robust and presented fewer errors. The amount of aliasing errors for the ordinary rotation form is about two times larger than for De-aliased XYZ rotational scheme and common skew-symmetric form. From a practical point of view, the choice between the skew-symmetric form and de-aliased rotational method with the same accuracy reduces to the last one if final decision rests on the amount of computational time.

The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension

Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.

High-order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates

In this paper, we discuss high-order finite difference weighted essentially non-oscillatory schemes, coupled with total variation diminishing (TVD) Runge–Kutta (RK) temporal integration, for solving the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintenance of third-order spatial/temporal accuracy when the limiters are applied to a third-order finite difference scheme and third-order TVD-RK time discretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth-order accuracy.

Thermodynamics of the one-dimensional parallel Kawasaki model: exact solution and mean-field approximations.

The adsorption isotherm for the recently proposed parallel Kawasaki (PK) lattice-gas model [Phys. Rev. E 88, 062144 (2013)] is calculated exactly in one dimension. To do so, a third-order difference equation for the grand-canonical partition function is derived and solved analytically. In the present version of the PK model, the attraction and repulsion effects between two neighboring particles and between a particle and a neighboring empty site are ruled, respectively, by the dimensionless parameters ϕ and θ. We discuss the inflections induced in the isotherms by situations of high repulsion, the role played by finite lattice sizes in the emergence of substeps, and the adequacy of the two most widely used mean-field approximations in lattice gases, namely, the Bragg-Williams and the Bethe-Peierls approximations.

New Oscillation Criteria for Third-Order Nonlinear Mixed Neutral Difference Equations

Some new oscillation criteria are established for a third-order nonlinear mixed neutral difference equation. Our results improve and extend some known results in the literature. Several examples are given to illustrate the importance of the results.