Proposed Difference Scheme Research Articles

Modelling of suspended particles motion in channel

The paper considers the application of the scheme based on linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error. The proposed difference scheme has shown its effectiveness in solving diffusion-convection problems in the range of grid Peclet numbers 2 ≥ Pe ≥ 20, which allows us to apply this class of schemes for the numerical solution of computational Oceanology problems. The problem of soil dumping was considered as a practical example of application of the proposed schemes.

An efficient computational approach for two-dimensional variant of nonlinear-dispersive model of shallow water wave

A three-level linearized difference scheme for two-dimensional dispersive shallow water wave that is governed by the Rosenau-RLW equation is considered. It is proved that the proposed difference scheme is conservative, uniquely solvable and unconditionally convergent. The convergence order in maximum norm is $$O(\tau ^2+h_1^2+h_2^2)$$, where $$\tau$$ is the temporal grid size and $$h_1, h_2$$ are spatial grid sizes in the x- and y-directions, respectively. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and other existing method. Comparison reveals that our method improves the accuracy of the space and time direction and shortens computation time largely.

The difference schemes for solving singularly perturbed three-point boundary value problem

In this paper, we propose and analyze numerical treatment for a singularly perturbed convection–diffusion boundary value problem with nonlocal condition. First, the boundary layer behavior of the exact solution and its first derivative have been estimated. Then we construct a finite difference scheme on a uniform mesh. We prove the uniform convergence of the proposed difference scheme and give an error estimate. We also present numerical examples, which demonstrate computational efficiency of the proposed method.

Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schr\xf6dinger equations

To improve the computing efficiency, a fourth-order difference scheme is proposed and a fast algorithm is designed to simulate the nonlinear fractional Schrödinger (FNLS) equation oriented from the fractional quantum mechanics. The numerical analysis and experiments conducted in this article show that the proposed difference scheme has the optimal second-order and fourth-order convergence rates in time and space respectively, reduces its computation cost to mathcal{O}(Mlog M), and recognizes accurately its physical feature of FNLS such as the mass balance.

Meshless upwind local radial basis function-finite difference technique to simulate the time- fractional distributed-order advection–diffusion equation

The main objective in this paper is to propose an efficient numerical formulation for solving the time-fractional distributed-order advection–diffusion equation. First, the distributed-order term has been approximated by the Gauss quadrature rule. In the next, a finite difference approach is applied to approximate the temporal variable with convergence order $$\mathcal{O}(\tau ^{2-\alpha })$$ as $$0<\alpha <1$$ . Finally, to discrete the spacial dimension, an upwind local radial basis function-finite difference idea has been employed. In the numerical investigation, the effect of the advection coefficient has been studied in terms of accuracy and stability of the proposed difference scheme. At the end, two examples are studied to approve the impact and ability of the numerical procedure.

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative**Project supported by the National Natural Science Foundation of China (Grant No. 11471262) and Henan University of Technology High-level Talents Fund, China (Grant No. 2018BS039).

The fractional Feynman–Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman–Kac equations, where the non-local time–space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman–Kac equation. The proposed difference schemes have the q-th (q = 1,2,3,4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman–Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

Uniform error estimates of fourth-order conservative linearized difference scheme for a mathematical model for long wave

This article is devoted to the study of high-order conservative linearized finite difference scheme for a model of nonlinear dispersive equation: regularized long-wave-Korteweg–de Vries equation. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergence of the difference scheme is proved by using the energy method to be of fourth-order in space and second-order in time in the discrete -norm. An application on the regularized long-wave equation as well for the modified regularized long-wave equation are discussed numerically in detail. 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. The temporal evaluation of a Maxwellian initial pulse is then studied for both equations. Some numerical results and comparisons with other existing methods in the literature are presented. The numerical results show that the linearized difference scheme of this article improves the accuracy of the space and time direction and shortens computation time largely.

О разностных схемах кабаре и крест

Для решения задачи переноса в статье предложено использовать схему, построенную на основе линейной комбинации разностной схемы кабаре (англ. Upwind Leapfrog) и крест (англ. Standard Leapfrog) с весовыми коэффициентами, полученными в результате минимизации погрешности аппроксимации. Проведено сравнение расчетов для задачи переноса на основе предложенной схемы с результатами, полученными с использованием схемы, построенной на основе линейной комбинации схемы с центральными разностями и схемы кабаре, и двухпараметрической разностной схемы третьего порядка точности. In order to solve the transfer problem, it is proposed to use the scheme based on a linear combination of the upwind and standard leapfrog difference schemes with weighting coefficients obtained by minimizing the approximation error. The estimate of the approximation error of the proposed difference scheme shows that, for small Courant numbers, this scheme whose approximation error is $O(ch^2)$, where the constant $c$ is significantly less than unity, is preferable to use than the original upwind and standard leapfrog schemes whose approximation errors are $O(h^2)$. The numerical results for the transfer problem based on the proposed scheme are compared with the results obtained using the following schemes: (i) the scheme based on a linear combination of the standard leapfrog scheme and the upwind leapfrog sscheme and (ii) the two-parameter difference scheme of the third order of accuracy.

Computation of Aerodynamic and Acoustic Characteristics of NACA0012 Airfoil Using the Zonal RANS–IDDES Approach

The presented results of computations of the aerodynamic and acoustic characteristics of an isolated symmetric NACA0012 airfoil with a blunt trailing edge in a subsonic homogeneous viscous perfect gas flow at zero angle of attack are obtained using the MP5 modified finite-difference scheme proposed in this study. The computations are accomplished within the zonal RANS–IDDES approach engaging the basic RANS Spalart–Allmaras model. A synthetic turbulence generator with a three-dimensional source is used to create turbulent content at the entrance to the IDDES domain. A series of model problems are considered, whose solutions allowed tuning the parameters of the proposed difference scheme and solving some methodological problems related to processing the simulation results.

Finite difference schemes for the two‐dimensional semilinear wave equation

In this article, two finite difference schemes for solving the semilinear wave equation are proposed. The unique solvability and the stability are discussed. The second‐order accuracy convergence in both time and space in the discrete H1‐norm for the two proposed difference schemes is proved. Numerical experiments are performed to support our theoretical results.

Об одной разностной схеме класса Кабаре повышенного порядка точности для решения уравнения переноса

Предложена новая разностная схема класса Кабаре повышенного порядка точности для решения скалярного уравнения переноса. Порядок аппроксимации разностной схемы равен четырем. Построено балансно-характеристическое представление схемы и приведены дисперсионные свойства. Для предложенной разностной схемы в сравнении с классической схемой Кабаре рассмотрены примеры решения уравнения переноса для гладкого и разрывного профиля. A new difference scheme of the Cabaret class with a higher order of accuracy for solving the scalar transport equation is proposed. The order of approximation of this difference scheme is equal to four. The balance-characteristic representation of the scheme is constructed and the dispersion properties are given. For the proposed difference scheme, a number of examples to solve the transport equation for smooth and discontinuous profiles are considered in comparison with the classical Cabaret scheme.

Finite difference scheme for multi-term variable-order fractional diffusion equation

In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order alpha(x,t) in(0,1) and the Riesz variable-order space fractional derivatives of order beta(x,t) in (0,1), gamma(x,t)in(1,2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes.

A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay

A numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the x–t plane is constructed. The presence of the perturbation parameter in the second-order space derivative gives rise to parabolic boundary layer(s) on one (or both) of the lateral side(s) of the rectangle. Thus the classical numerical methods on the uniform mesh are inadequate and fail to give good accuracy and results in large oscillations as the perturbation parameter approaches zero. To overcome this drawback a numerical method comprising the Crank–Nicolson finite difference method consisting of a midpoint upwind finite difference scheme on a fitted piecewise-uniform mesh of $$N\times M$$ elements condensing in the boundary layer region is constructed. A priori explicit bounds on the solution of the problem and its derivatives which are useful for the error analysis of the numerical method are established. To establish the parameter-uniform convergence of the proposed method an extensive amount of analysis is carried out. It is shown that the proposed difference scheme is second-order accurate in the temporal direction and the first-order (up to a logarithmic factor) accurate in the spatial direction. To validate the theoretical results, the method is applied to two test problems. The performance of the method is demonstrated by calculating the maximum absolute errors and experimental orders of convergence. Since the exact solutions of the test problems are not known, the maximum absolute errors are obtained by using double mesh principle. The numerical results show that the proposed method is simply applicable, accurate, efficient and robust.

Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems

The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto–Sivashinsky equation and extended Fisher–Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.

Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 &lt; α &lt; 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

The difference scheme for the two-dimensional convection-diffusion problem for large peclet numbers

The purpose of this work is the development of a difference scheme for the solution of convection-diffusion problem at high Peclet numbers (Pe&gt;2). In accordance with this purpose the following problems were solved: difference scheme for convection is built, comparison with the existing schemes is carried out; conditions for stability of the proposed difference scheme are obtained. Solutions of the convection-diffusion equation on the basis of the proposed difference scheme at various Peclet numbers are obtained.

The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for Solving the Time Multi-term and Distributed-Order Fractional Sub-diffusion Equations

In this article, a special point is found for the interpolation approximation of the linear combination of multi-term fractional derivatives. The derived numerical differentiation formula can achieve at least second order accuracy. Then the formula is used to numerically solve the time multi-term and distributed-order fractional sub-diffusion equations. Several unconditionally stable and convergent difference schemes are presented. The stability and convergence of the difference schemes are discussed. Some numerical examples are provided to show the efficiency of the proposed difference schemes.

Stability of Difference Splitting Schemes for Convection Diffusion Equation

We consider numerical modeling of the propagation of pollution in the air on the basis of geometrical splitting method for three-dimensional nonstationary convection diffusion equations. Splitting difference schemes in the form of explicit computing schemes are proposed to solve the obtained one-dimensional problems. The approximation, monotonicity, and stability of the proposed difference schemes are investigated.

On a highly accurate approximation of the first and pure second derivatives of the Laplace equation in a rectangular parallelpiped

We propose and justify difference schemes for the approximation of the first and pure second derivatives of a solution of the Dirichlet problem in a rectangular parallelepiped. The boundary values on the faces of the parallelepiped are supposed to have six derivatives satisfying the Holder condition, to be continuous on the edges, and to have second- and fourth-order derivatives satisfying the compatibility conditions resulting from the Laplace equation. We prove that the solutions of the proposed difference schemes converge uniformly on the cubic grid of order $O(h^{4})$ , where h is a grid step. Numerical experiments are presented to illustrate and support the analysis made.

Fourth order finite difference schemes for time–space fractional sub-diffusion equations

In this paper, we devote to the study of high order finite difference schemes for one- and two-dimensional time–space fractional sub-diffusion equations. A fourth order finite difference scheme is invoked for the spatial fractional derivatives, and the L1 approximation is applied to the temporal fractional parts. For the two-dimensional case, an alternating direction implicit scheme based on L1 approximation is proposed. The stability and convergence of the proposed methods are studied. Numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes.