Methods Of High-order Accuracy Research Articles

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.

A Displacement Discontinuity Method of High-Order Accuracy in Fracture Mechanics

In this paper the displacement discontinuity method of high-order accuracy and its application to the problems of fracture mechanics are considered. In common practice of applications of boundary element methods, the methods with the piecewise-constant function of boundary displacement are often used. Their advantage against other algorithms is the simplicity of calculation scheme with a rather good accuracy of the solution at the points of region distant from the boundary. In the fracture mechanics (with lines of surfaces of discontinuity of the displacement field), it is required to describe the stress behavior in the proximity of the crack edges with the highest accuracy possible, which leads to necessity of increasing the degree of accuracy of the used numerical methods. It is shown that the methods with high-order continuity of displacements at the boundary proposed in this work substantially improve the accuracy of computation of displacement and stress fields in the neighborhood of crack edges within the region.

The least squares collocation method for the biharmonic equation in irregular and multiply-connected domains

This paper reports new h-and p-versions of the least squares collocation method of high-order accuracy proposed and implemented for solving boundary value problems for the biharmonic equation in irregular and multiply-connected domains. This paper shows that approximate solutions obtained by the least squares collocation method converge with high order and agree with analytical solutions of test problems with high degree of accuracy. There has been a comparison made for the results achieved in this study and results of other authors who used finite difference and spectral methods.

Finite Element Method of High-Order Accuracy for Solving Two Dimensional Elliptic Boundary-Value Problems of Two and Three Identical Atoms in a Line

We considered models of three identical atoms in a line with molecular pair interactions and diatomic molecule scattered by an atom or tunneling through potential barriers. The models are formulated as 2D elliptic boundary-value problems (BVPs) in the Jacobi and polar coordinates. The BVP in Jacobi coordinates solved by finite element method of high-order accuracy for discrete spectrums of models under consideration. To solve the scattering problems the BVP in polar coordinates are reduced by means of Kantorovich method to a system of second-order ordinary differential equations with respect to the radial variable using the expansion of the desired solutions in the set of angular basis functions that depend on the radial variable as a parameter. The efficiency of the elaborated method, algorithms and programs is demonstrated by benchmark calculations of the resonance scattering, metastable and bound states of the considered models and also by a comparison of results for bound states of the three atomic system in the framework of direct solving the BVP by FEM and Kantorovich reduction.

Versions of the Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Convex Quadrangular Domains

The new versions of the collocations and least residuals (CLR) method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for PDE in the convex quadrangular domains. Their implementation and numerical experiments are performed by the examples of solving the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. Differential problems are projected into the space of fourth-degree polynomials by the CLR method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLR method are implemented on the grids, which are constructed by two different ways. In the first version, a “quasiregular” grid is constructed in the domain, the extreme lines of this grid coincide with the boundaries of the domain. In the second version, the domain is initially covered by a regular grid with rectangular cells. Herewith, the collocation and matching points that are situated outside the domain are used for approximation of the differential equations in the boundary cells that had been crossed by the boundary. In addition the “small” irregular triangular cells that had been cut off by the domain boundary from rectangular cells of the initial regular grid are joined to adjacent quadrangular cells. This technique allowed to essentially reduce the conditionality of the system of linear algebraic equations of the approximate problem in comparison with the case when small irregular cells together with other cells were used as independent ones for constructing an approximate solution of the problem. It is shown that the approximate solution of problems converges with high order and matches with high accuracy with the analytical solution of the test problems in the case of the known solution in numerical experiments on the convergence of the solution of various problems on a sequence of grids.

Exact three-point difference scheme for singular nonlinear boundary value problems

Exact three-point difference scheme on an irregular grid for numerically solving boundary value problems for second order nonlinear ordinary differential equations with a singularity of the first kind is constructed. We have proved the existence and uniqueness of the solution of this scheme as well as have shown the convergence of the associated iteration method. In order to determine the coefficients and the right-hand side of the exact difference scheme at an arbitrary node of the grid, some auxiliary initial value problems on a small interval around this node must be solved. If these initial value problems are solved numerically by a one-step method of high-order accuracy, truncated three-point difference schemes of high-order accuracy results. The effectiveness of proposed difference schemes is illustrated by a numerical example.

A finite-difference method of high-order accuracy for the solution of transient nonlinear diffusive–convective problem in three dimensions

This paper presents an efficient technique of linearization of the nonlinear convective terms present in engineering problems involving heat and mass transfer and fluid mechanics. From two numerical applications, this technique with a method of high-order finite differences is validated by numerical solution of transient nonlinear diffusive–convective problem in three dimensions.

A Finite Difference Method of High Order Accuracy for the Solution of Two-Point Boundary Value Problems

We present a new high order finite difference method for second order differential equation y''=f(x,y) subject to boundary conditions y(a)=alpha and y(b)= beta. The method is based on rational function approximation and its development is based on power series expansions. Under appropriate conditions, local truncation error calculated and order of method estimated six. Our finite difference method leads to nonlinear system of equations. Numerical examples are given to illustrate the effectiveness, efficiency and high order accuracy of the method.

Methods of high level exact solution of Cauchy’s task for Volter’s unlinear integro-partial differential equations

The new applications of continued fraction to the development of numerical methods for the solution of integrodifferential equations are considered. The numerical methods of high-order accuracy for the solution of initial-value problems in Volterra integro-differential equations are proposed.

A novel numerical method of high‐order accuracy for flow in unsaturated porous media

SUMMARYWe construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three‐stage predictor–corrector procedure. First, a high‐order weighted essentially non‐oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global space–time DG FEM is used to obtain a scheme without the parabolic time‐step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a Newton–Krylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one‐step scheme, on the basis of the solution calculated previously in the space–time predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time. Copyright © 2011 John Wiley & Sons, Ltd.

Benchmark Kantorovich calculations for three particles on a line

A Kantorovich approach is used to solve for the eigenvalue and the scattering properties associated with a multi-dimensional Schrödinger equation. It is developed within the framework of a conventional finite element representation of solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated in the case of an exactly solvable ‘benchmark’ model of three identical particles on a line, with zero-range attractive pair potentials and below the three-body threshold. In this model all the ‘effective’ potentials, and ‘coupling matrix elements’, of the set of resulting close-coupling radial equations, are calculated using analytical formulae. Variational formulations are developed for both the bound-state energy and the elastic scattering problem. The corresponding numerical schemes are devised using a finite element method of high order accuracy.

Direct Numerical Simulation of Turbulent Flow in a Wavy Channel. 1st Report. Characteristics of Flow Field.

A direct numerical simulation (DNS) of a turbulent flow in a wavy channel was carried out with consistent finite-difference method of high-order accuracy. The application of the database generated by the DNS is suited not only for study of the structure of turbulent flow but also for the design and testing of turbulence models in more practical fields. The wavy channel causes periodic pressure gradient, successively acceleration and deceleration of streamwise velocity, and separating region. The scale of separating region depends on the amplitude of wavy wall. Some characteristics of the flow are shown, for example the momentum distribution. It is also demonstrated that the flow is characterized by the simple linear approximation

Derivative error bounds for Lagrange interpolation: An extension of Cauchy's bound for the error of Lagrange interpolation

It is shown that for any n + 1 times continuously differentiable function f and any choice of n + 1 knots, the Lagrange interpolation polynomial L of degree n satisfies ∥f (n) − L (n)∥ ⩽ ∥ω (n)∥ (n + 1)! ∥tf (n + 1)∥ , where ∥ ∥ denotes the supremum norm. Further, this bound is the best possible. Applications of the above bound to the differencing formula are suggested. It is also shown that for j = 1, 2, …, n − 1, ∥f (j) − L (j)∥ ⩽ ∥ω (j)∥ j!(n + 1 − j)! ∥tf (n + 1)∥ . This formula may be considered as a generalization of a formula due to Ciarlet, Schultz, and Varga (Numerical methods of high-order accuracy, Numer. Math. 9 (1967), 394–430) and may be compared to the conjectured best bound ∥f (j) − L (j)∥ ⩽ ∥ω (j)∥ (n + 1)! ∥tf (n + 1)∥ .

High order accurate vortex methods with explicit velocity kernels

Vortex methods of high order accuracy are developed for inviscid, incompressible fluid flow in two or three space dimensions. The velocity kernels are smooth functions given by simple, explicit formulas. Numerical results are given for test problems with exact solutions in two dimensions. It is found that the higher order methods yield a considerably more accurate representation of the velocity field than those of lower order for moderate integration times. On the other hand, the velocity field computed by the point vortex method has very poor accuracy at locations other than the particle trajectories.

Finite Element Methods of High-Order Accuracy for Singular Two-Point Boundary Value Problems with Nonsmooth Solutions

In this paper we analyze several finite element methods for singular two-point boundary value problems\[ - (x^\sigma u')' + qu = f,\quad 0 < x < 1.\] Here $\sigma \in [0,1)$, and appropriate boundary conditions are imposed. The solution can be approximated by splines on a nonuniform (“$\beta $ -graded”) mesh. Error bounds of optimal order are proved, and upper and lower bounds on the extent to which the mesh must be graded are obtained. We also consider approximating the solution by functions of the form $x^{ - \sigma } s(x),s(x)$ a spline. Error bounds and numerical results for these “weighted splines” indicate hat they are very efficient. For a third subspace, known error bounds are improved by using a mildly graded mesh.

An implicit, compact, finite difference method to solve hyperbolic equations

At present most approximate (discrete) solutions of time dependent hyperbolic equations are obtained by explicit finite difference methods, where the maximal allowable time step is given by a condition of numerical stability (i.e., the CFL condition). This report contains the development and the analysis of an implicit method of high order accuracy which is unconditionally stable, thus allowing to progress much faster in time. Furthermore, the presence of additional artificial boundary conditions does have an influence on the accuracy of the solution but not on the stability of the method. The numerical scheme has been checked considering two examples: the solution of the (linear) wave equation and the non linear Euler equations of fluid dynamics.

A method of high-order accuracy for the numerical integration of boundary value problems

A finite-difference method for the integration of the linear two-point boundary value problem $$y'' = f(x)y + g(x), y(a) = y_a , y(b) = y_b $$ is constructed. It uses values off′,f″,g′,g″ at the grid points to obtainO(h8) global error, and allows strict global error bounds, while needing only the solution of a tridiagonal system of equations. Numerical examples are presented to demonstrate the practical usefulness of our method.

Numerical Methods of High Order Accuracy for One-Dimensional Diffusion Equation

The classical Rayleigh-Ritz procedure is applied to the variational formulation of the one-dimensional diffusion equation. By minimizing the corresponding functional over finite dimensional piecewi...

The Rate of Convergence for the Finite Element Method

The character of the proper refinement of the elements (mesh) around the boundary is studied. It is shown that it is possible to obtain the highest (optimal) rate of convergence by this refinement.

An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

For a differential equationdx/dt=f(t, x) withf t (t, x),f x (t, x) computable, the author presents a new one-step method of high-order accuracy. A rule of controlling the mesh size is given and the method is compared with the Runge-Kutta method in two numerical examples.