Difference Schemes Of Order Research Articles

Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations

In this paper, we construct a high order difference scheme for two-dimensional semilinear fractional sub-diffusion equations at first. To reduce the computation time, an efficient time two-grid algorithm is then proposed. The global convergence order of the two-grid scheme reaches O(τF2+τC4+h12+h22), where τF and τC represent the time-step sizes on the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence of the scheme are carefully studied by some skills. At last, the theoretical statements are verified by numerical experiments.

Finite difference schemes of the 4th order of approximation for Maxwell's equations

Finite difference schemes of the 4th order of approximation for Maxwell's equations

DIFFERENCE SCHEME METHOD FOR THE NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS

In this study, wave equations with initial boundary conditions have been studied. The general form of the wave equation has been derived. The first order and second order difference schemes were established for the presented IBVP. The stability of the difference schemes has been guaranteed. The approximation solution of the problem was achieved by using finite difference methods. Two different examples are provided. A comparison between the exact and approximation solution has been carried out. Absolute errors of the problem have been presented by using MATLAB software. Moreover, the comparison shows that the second order difference scheme is a more accurate result than the first order. It is shown that the results of the comparison guaranty the reliability and accuracy of the presented method.

Second order difference scheme for singularly perturbed boundary turning point problems

A singularly perturbed convection diffusion equation with boundary turning point is considered in this paper. A higher order method on piecewise uniform Shishkin mesh is suggested to solve this problem. We prove that this method is of order $O(N^{-2}(ln N)^2)$. Numerical results are given which validate the analytical results.

Stable numerical schemes for time-fractional diffusion equation with generalized memory kernel

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order ( 2 − α ) on a non-uniform grid where 0 < α < 1 . Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L 2 -norm and L ∞ -norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results.

Numerical Solution of the Wave Propagation Problem in a Plate

In this work, we use the phase velocity method in combination with finite element method to compute the dispersion curve for phase velocity of an ultrasonic pulse traveling in a thin isotropic plate. This method is based on the numerical solution of the wave propagation equations for several selected frequencies. To solve these equations, a second order difference scheme is used to discretize the temporal variable, while spatial variables are discretized using the finite element method. The variational formulation of the problem corresponding to a fixed value of time is obtained and the existence and uniqueness of the solution is proved. A priori error estimates in the energy norm and in the [Formula: see text] norm are also obtained. The open software FreeFem++ is used with quadratic triangular elements to compute the displacements. Numerical experiments show that the velocities computed from the approximated displacements for different frequency values are in good agreement with analytical dispersion curve. This confirms that the proposed symbiosis between finite element and phase velocity method is suitable for computing dispersion curves in more general wave propagation problems, where the geometry is complex and the material is anisotropic.

High accuracy analysis of Galerkin finite element method for Klein–Gordon–Zakharov equations

• We propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov equations with power law nonlinearity. • we use the combination of the interpolation and Ritz projection technique, which can reduce the regularity of exact solution. • We study the optimal and superconvergent error estimate results both theoretically and numerically. • We discuss the extensions of our scheme to more general finite elements. The main aim of this paper is to propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, and to give the error estimations of approximate solutions about the electronic fast time scale component q and the ion density deviation r . In which, the bilinear element is used for spatial discretization, and a second order difference scheme is implemented for temporal discretization. Moreover, by use of the combination of the interpolation and Ritz projection technique, and the interpolated postprocessing approach, for weaker regularity requirements of the exact solution, the superclose and global superconvergence estimations of q in H 1 − norm are deduced. At the same time, the superconvergence of the auxiliary variable φ ( − Δ φ = r t ) in H 1 − norm and optimal error estimation of r in L 2 − norm are derived. Meanwhile, we also discuss the extensions of our scheme to more general finite elements. Finally, the numerical experiments are provided to confirm the validity of the theoretical analysis.

Solving the Sturm-Liouville problem by three-point difference schemes of high order of accuracy

For the solving Sturm-Liouville problem, three-point difference schemes of high order of accuracy on a nonuniform grid are constructed. It is shown that the coefficients of these schemes are expressed in terms of solutions of two auxiliary initial value problems. An estimate of the accuracy of three-point difference schemes is obtained and an iterative Newton method is proposed to determine their solution. Numerical experiments confirm theoretical conclusions.

Compact Difference Schemes on a Three-Point Stencil for Second-Order Hyperbolic Equations

We consider compact difference schemes of approximation order 4+2 on a three-point spatial stencil for theKlein–Gordon equations with constant and variable coefficients. New compact schemes areproposed for one type of second-order quasilinear hyperbolic equations. In the case of constantcoefficients, we prove the strong stability of the difference solution under small perturbations ofthe initial conditions, the right-hand side, and the coefficients of the equation. A priori estimatesare obtained for the stability and convergence of the difference solution in strong mesh norms.

Mathematical modeling of zooplankton productivity in the Azov Sea in summer on a high-performance computer system

The work is devoted to modeling of eurythermal and stenothermal zooplankton populations productivity in the Azov Sea in summer. The discretization of the problem was carried out using difference schemes of a higher order of accuracy, taking into account the filling of the calculated cells. The grid equations obtained as a result of the proposed mathematical model of biological kinetics discretization were solved by a modified adaptive variable-triangular method, which has a higher convergence rate in comparison with the existing methods of the variational type. A parallel algorithm for solving the problem on a high-performance computing system was developed, which made it possible to significantly reduce the time of the numerical solution of the problem. With the help of mathematical modeling, the analysis of various ways of development of the Azov Sea ecosystem was carried out, the influence of the gelatinous invasive on the production and de-struction processes of phyto- and zooplankton was researched. The developed software module, oriented to a multiprocessor computing system, can be effectively used for predictive modeling of interrelated hydrodynamic and biogeochemical processes in a shallow water body in summer.

Numerical Solution of a Singularly Perturbed Fredholm Integro Differential Equation with Robin Boundary Condition

In this paper, we deal with singularly perturbed Fredholm integro differential equation (SPFIDE) with mixed boundary conditions. By using interpolating quadrature rules and exponential basis function, fitted second order difference scheme has been constructed on a Shishkin mesh. The stability and convergence of the difference scheme have been analyzed in the discrete maximum norm. Some numerical examples have been solved and numerical outcomes are obtained.

High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes

In this article, we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes. The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high. We can obtain sparse matrices by applying compact schemes. In this article, we compare compact and exponential finite difference schemes of fourth order. The numerical solutions are calculated in quadruple precision (Real * 16 or extended precision) in FORTRAN language, and iteratively obtained until reaching the round-off error magnitude around 1.0E −32. This procedure is performed to ensure that there is no iteration error. The Repeated Richardson Extrapolation (RRE) method combines numerical solutions in different grids, determining higher orders of accuracy. The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth, eighth, and tenth order of precision. The multigrid Full Approximation Scheme (FAS) is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.

Three-point difference schemes of high accuracy order for Sturm–Liouville problem

Three-point difference schemes of high accuracy order for Sturm–Liouville problem

Second order difference schemes for time-fractional KdV–Burgers’ equation with initial singularity

In this paper, we study the numerical method for time-fractional KdV–Burgers’ equation with initial singularity. The famous L 2 - 1 σ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments.

Using matrix stability for variable telegraph partial differential equation

The variable telegraph partial differential equation depend on initial boundary value problem has been studied. The coefficient constant time-space telegraph partial differential equation is obtained from the variable telegraph partial differential equation throughout using Cauchy-Euler formula. The first and second order difference schemes were constructed for both of coefficient constant time-space and variable time-space telegraph partial differential equation. Matrix stability method is used to prove stability of difference schemes for the variable and coefficient telegraph partial differential equation. The variable telegraph partial differential equation and the constant coefficient time-space telegraph partial differential equation are compared with the exact solution. Finally, approximation solution has been found for both equations. The error analysis table presents the obtained numerical results.

On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations

The initial value problem for the third order delay differential equation in a Hilbert space with an unbounded operator is investigated. The absolute stable three-step difference scheme of a first order of accuracy is constructed and analyzed. This difference scheme is built on the Taylor’s decomposition method on three and two points. The theorem on the stability of the presented difference scheme is proven. In practice, stability estimates for the solutions of three-step difference schemes for different types of delay partial differential equations are obtained. Finally, in order to ensure the coincidence between experimental and theoretical results and to clarify how efficient the proposed scheme is, some numerical experiments are tested.

Multistep schemes for one and two dimensional electromagnetic wave models based on fractional derivative approximation

In this article, multistep finite difference schemes are developed for solving one dimensional (1D) and two dimensional (2D) fractional differential models of electromagnetic waves (FDMEWs) arising from dielectric media which contain both initial and Dirichlet boundary conditions. The Caputo’s fractional derivatives in time are discretized by a difference scheme of order O ( τ 3 − α ) & O ( τ 3 − β ) , 1 < β < α < 2 , and the Laplacian operator is approximated by central difference discretization. The proposed multistep schemes transform the FDMEWs into the tridiagonal system for 1D case and pentagonal system for 2D case. Theoretical unconditional stability, convergence analysis and error bounds are investigated. For 1D FDMEWs, accuracy of order O ( τ 3 − α + τ 3 − β + h 2 ) and for 2D FDMEWs, accuracy of order O ( τ 3 − α + τ 3 − β + h 1 2 + h 2 2 ) are investigated, where 1 < β < α < 2 . Several test examples are included to verify the reliability and computational efficiency of the proposed schemes which support our theoretical findings for both 1D and 2D cases.

Comparison of Gradient Approximation Methods in Schemes Designed for Scale-Resolving Simulations

Various methods for the accurate approximation of the gradients entering the diffusion fluxes are considered. Linear combinations of the 2nd order difference schemes for a non-uniform mesh that transform into 4th order schemes in the uniform case are investigated. We also consider 3rd and 4th order schemes for approximating gradients on a non-uniform mesh in the normal and tangential directions to the cell face, respectively, based on Lagrange polynomials. The initial testing is carried out on one-dimensional functions: a smooth Gauss function and a piecewise linear function. Next, the schemes are applied in direct numerical simulation of the Taylor–Green vortex.

An efficient numerical method for solving Benjamin–Bona–Mahony–Burgers equation using difference scheme

ABSTRACTIn this paper, we propose a numerical solution of the Benjamin–Bona–Mahony–Burgers equation (BBMB) based on finite difference scheme of sixth order. It is proved that the scheme is convergent. The existence and stability analysis of numerical solutions are discussed. The method is applied to several examples and accuracy of the method is tested in terms of and error norms. Furthermore, the numerical results have been compared with some other methods.

Numerical calculation of elements of thin-walled structures under alternating loading taking into account damage

The statement of the problem and the algorithms for calculating thin-walled rods of arbitrary section under spatially variable loading based on the theory of small elastoplastic deformations and the refined theory of rods are presented. Using the variational Lagrange principle, mathematical models are developed for the deformation and damage of thin-walled rods in a cylindrical coordinate system. A system of differential equilibrium equations for a rod with spatially variable loading in vector form is obtained. To solve the boundary-value problem, the central difference scheme of the second order of accuracy and the matrix sweep method are used. An example of calculation is given.