Accurate Difference Scheme Research Articles

Source identification problems for the neutron transport equations

In this study, the time-dependent source identification problem for the two-dimensional neutron transport equation was studied. For the approximate solution of this problem a first order of accuracy difference scheme was presented. Stability estimates for the solution of these differential and difference problems were established. Numerical results were given.

A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations

In the present paper, we construct a first order of accuracy difference scheme for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise dependent source term. The unique solvability of constructed difference scheme and the stability estimates for its solution were obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

Absolutely stable difference scheme for the delay partial differential equation with involution and Robin boundary condition

This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements.

Stable difference schemes for hyperbolic–parabolic equations with unknown parameter

In the present paper, we study the first and second order of accuracy difference schemes for the approximate solution of the inverse problem for hyperbolic–parabolic equations with unknown time-independent source term. The unique solvability of constructed difference schemes and the stability estimates for their solutions are obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems.

Bounded Solutions of Semi-Linear Parabolic Differential Equations with Unbounded Delay Terms

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is established. The application of the main theorem to four different semi-linear delay parabolic differential equations is presented. The first- and second-order accuracy difference schemes (FSADSs) for the solution of a one-dimensional semi-linear time-delay parabolic equation are considered. The new desired numerical results of this paper and their discussion are presented.

NUMERICAL SIMULATION OF A FLUID FLOW IN THE CYLINDRICAL DUCT CONTAINING TWO DIAPHRAGMS WITH ORIFICES OF DIFFERENT DIAMETERS

The flow of a viscous incompressible fluid in a cylindrical duct with two serial diaphragms with orifices of different diameters was studied based on the numerical solution of the unsteady Navier-Stokes equations. The solution algorithm was based on the finite volume method using second-order accurate difference schemes in both space and time. The TVD (Total-Variation Diminishing) form of a central-difference scheme with a flux limiter was used for the interpolation of the convective terms. The combined evaluation of the velocity and pressure fields was carried out using the PISO (Pressure Implicit Split Operator) procedure. The problem was solved using OpenFOAM (Open-source Field Operation And Manipulation) open-source toolkit libraries using the computing power of the cluster supercomputer of the V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. It was shown that within a certain range of the diameter ratio of the orifices of the diaphragms, a circulation movement is established in the cavity between the diaphragms. The boundary layer breaks off from the surface of the first diaphragm and forms an annular shear layer. When approaching the second diaphragm, a sequence of ring vortices is formed in it, which interact with the surface of the diaphragm and lead to the emergence of tonal sound. When the ratio of the orifice diameter of the second diaphragm to the first decreases, the share of the jet’s kinetic energy, which participates in the circulation in the middle of the cavity between the diaphragms, increases. As a result, the amplitude of velocity fluctuations in the orifice of the second diaphragm decreases. When the ratio of the diameters of the orifices increases, the share of energy involved in the circulation decreases, and when a certain value of the ratio is reached, the interaction between the vortices in the shear layer and the surface of the diaphragm does not occur. As a result, the excitation of the tonal sound stops. The Strouhal number of self-oscillations practically does not change when the diameter ratio of the orifices changes. During the calculations, two different modes of self-oscillation were obtained, which is consistent with previous works.

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.

Spurious currents suppression by accurate difference schemes in multiphase lattice Boltzmann method

Spurious currents are often observed near curved interfaces in multiphase simulations based on diffuse interface methods. These unphysical phenomena negatively affect both computational accuracy and stability. In this paper, the origin and suppression of spurious currents are investigated using the multiphase lattice Boltzmann method driven by a chemical potential. Both the difference error and insufficient isotropy of the discrete gradient operator give rise to directional deviations in the nonideal force, leading to spurious currents. Nevertheless, a high-order finite difference scheme produces far more accurate results than a high-order isotropic difference scheme. We compare several finite difference schemes that have different levels of accuracy and resolution. When a large proportional coefficient is used, the transition region is narrow and steep, and the resolution of the finite difference scheme provides a better indication of the computational accuracy than the formal accuracy. Conversely, for a small proportional coefficient, the transition region is wide and gentle, and the formal accuracy of the finite difference gives a better indication of the computational accuracy than the resolution. Numerical simulations show that the spurious currents calculated in the three-dimensional situation are highly consistent with those in two-dimensional simulations; in particular, the two-phase coexistence densities calculated by the high-order accuracy finite difference scheme are in excellent agreement with the theoretical predictions of the Maxwell equal-area construction down to a reduced temperature of 0.2.

Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems

This is a discussion on the second-order accuracy difference schemes for approximate solution of the integral-type time-nonlocal parabolic problems. The theorems on the stability of r-modified Crank-Nicolson difference schemes and second-order accuracy implicit difference scheme for approximate solution of the integral-type time-nonlocal parabolic problems in a Hilbert space with self-adjoint positive definite operator are established. In practice, stability estimates for the solutions of the second-order accuracy in t difference schemes for the one and multidimensional time-nonlocal parabolic problems are obtained. Numerical results are given.

An efficient pure meshless method for phase separation dominated by time-fractional Cahn–Hilliard equations

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution.

A Note on Stability of Parabolic Difference Equations on Torus

The present article investigates nonlocal boundary value problems for parabolic equations of reverse type on torus. The first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problems for parabolic equations on circle and torus are presented. For the solutions of the difference scheme, the stability estimates and coercivity estimates in various Hölder norms are established. Furthermore, theoretical results are supported by numerical experiments.

Highly accurate difference schemes for time‐nonlocal Schrodinger type problems

AbstractIn this study, time‐nonlocal Schrodinger type problems are investigated. Single step absolute stable highly accurate difference schemes for the numerical solution of these problems are presented and theorems on the stability of these difference schemes are established. Numerical illustrations in test examples for the third and fourth order of accuracy difference schemes are presented.

A fourth‐order accurate numerical method for the distributed‐order Riesz space fractional diffusion equation

AbstractIn this article, a novel fourth‐order accurate difference method is derived for the distributed‐order Riesz space fractional diffusion equation in one‐dimensional (1D) and two‐dimensional (2D) cases, respectively. First, the distributed integral terms are discretized by using the Simpson quadrature rule into the multi‐term Riesz space fractional diffusion equations. Then, a fourth‐order accurate difference scheme is presented to approximate the multi‐term Riesz fractional diffusion equations. Moreover, the proposed difference schemes are proved to be unconditionally stable and convergent in norm for both 1D and 2D cases. Finally, numerical experiments are given to verify the efficiency of the schemes.

High-fidelity robust and efficient finite difference algorithm for simulation of polymer-induced turbulence in cylindrical coordinates

A high-fidelity finite difference method for simulation of viscoelastic turbulence in cylindrical coordinates has been developed. In this algorithm, the evolution equation for the polymer conformation tensor is discretized by an improved Kurganov–Tadmor (KT) scheme, while the continuity and momentum equations are discretized by a well-established second-order accurate central difference scheme on staggered grids. To minimize the computational cost of the original KT scheme, two novel improvements have been implemented: (i) a sequential ranking strategy to rapidly select the slope-approximation in the original KT scheme, and (ii) a simple criterion to diagnose and in turn assure the positive definiteness of the conformation tensor that obviates the need for CPU intensive eigenvalue calculations. These modifications lead to an algorithm that is twice as fast as the original KT method. This in turn has enabled high-fidelity and efficient larger-scale simulations of viscoelastic turbulent flows in a parallel framework using 2D-domain decomposition, as evinced by benchmark results for two canonical viscoelastic flows, namely, the viscoelastic pipe flows (VPF) and the viscoelastic Taylor–Couette flows (VTCF). To that end, the fidelity and computational efficiency of this algorithm have been demonstrated in a broad range of polymer-induced flow dynamics, namely, elasto-inertial instability in the VPF, polymer-induced drag reduction at high R e in the VTCF, and purely elastic instability in low- R e VTCF. Overall, this algorithm is an efficient and high-fidelity scheme for the simulation of viscoelastic turbulence in a broad range of elastically and/or inertially dominated flows. • A new robust finite-difference method for viscoelastic turbulence is proposed. • High-fidelity simulations are performed using the proposed method. • Strategies to improve computational efficiency are proposed. • Comprehensive validations in cylindrical coordinates are carried out.

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral‐type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self‐adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral‐type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.

A new high-order accurate difference scheme for the Benjamin-Bona-Mahoney-Burgers (BBMB) equation

In this article, a high-order linearized difference scheme is presented for the periodic initial value problem of the Benjamin-Bona-Mahoney-Burgers (BBMB) equation. It is proved that the proposed scheme is uniquely solvable and unconditionally convergent, with convergence order of O(h4 + k2) in the L?-norm. An application on the regularised long wave is thoroughly studied numerically. Furthermore, interaction of solitary waves with different amplitudes is shown. The three invariants of the motion are evaluated to determine the conservation properties of the system. Numerical experiments including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of our difference scheme and to confirm the theoretical analysis.