Order conditions and symmetry for two-step hybrid methods

In this study of two-step hybrid methods for second-order equations of the type y″?=?f(y), we apply P-series [Hairer, E., Lubich, C. and Wanner, G. (2002). Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics.] to formalise the approach of Chan [Chan, R. P. K. (2002). Two-step hybrid methods. Internal Publication.] to the order conditions, and present two characterizations of symmetry. Although order conditions can be obtained through the classical theory for the Nyström methods, it is of interest to derive particular simpler formulas for the class of two-step hybrid methods in order to facilitate the search for high-order methods. Moreover, the approach proves useful in analysing the symmetry of the hybrid methods.     

Multidimensional ARKN methods for general oscillatory second-order initial value problems

Based on B-series theory, the order conditions of the multidimensional ARKN methods are presented for the general multi-frequency and multidimensional oscillatory second-order initial value problems by Wu et al. (2009). These multidimensional ARKN methods exactly integrate the multi-frequency and multidimensional unperturbed oscillators. In this paper, we pay attention to the analysis of the concrete multidimensional ARKN methods for the general multi-frequency oscillatory second-order initial value problems whose right-hand side functions depend on both  y$y$and  y^′$y^{\prime }$ (the class of physical problems which fall within its scope is broader). Numerical experiments are carried out to show that the new multidimensional ARKN methods are more efficient compared with some well-known methods for dealing with the oscillatory problems in the scientific literature.     

A partitioning strategy for explicit difference methods

Two partition shapes, slices and rectangles, are analyzed for the case of explicit difference methods on regular grids. The question is how to partition the grid in order to minimize the execution time on MIMD computers with distributed memory. The amount of computational work at the boundary points is assumed to be different from that the interior points. It is shown that a good strategy is to nevertheless do the partitioning as if the work was equal at all point. This strategy is nearly optimal, in the sense that it will in most cases give only a small reduction of the efficiency. At the same time it has the advantage of making it straightforward to compute the partitioning. The only situation where this strategy is not recommendable is when the work is much larger at the boundary points than in the interior part of the computational domain.     

A class of explicit preconditioned conjugate gradient methods for solving large finite element systems

A class of Explicit Preconditioned Conjugate Gradient (EPCG) methods for solving large sparse linear systems of algebraic equations resulting from the Finite Element discretization of Elliptic and Parabolic PDE's is introduced. The EPCG methods are based on explicit Approximate Inverse Matrix techniques and are particularly suitable for solving numerically initial/boundary-value problems on multiprocessor systems. The application of the new methods on 2D-linear boundary-value problems is discussed and numerical results are given.     

Finite termination and global monotonicity of Newton-type methods for solving hybrid piecewise linear systems

In the present paper, we give a detailed theoretical analysis for some Newton-type procedures for certain piecewise linear systems. Under rather general assumptions, the iterates are well defined and monotonically converge to the exact solution of the given systems. This procedure is shown to have a finite termination property, i.e. it converges to the exact solution in a finite number of steps.     

A class of explicit two-step superstable methods for second-order linear initial value problems

A class of explicit two-step superstable methods of fourth algebraic order for the numerical solution of second-order linear initial value problems is presented in this article. We need Taylor expansion at an internal grid point and collocation formulae for the derivatives of the solution to derive a method and then modify it into a class of methods having the desired stability properties. Computational results are presented to demonstrate the applicability of the methods to some standard problems.     

Study of general Taylor-like explicit methods in solving stiff ordinary differential equations

In this paper, we present a further study of Taylor-like explicit methods in solving stiff ordinary differential equations. We derive the general form for Taylor-like explicit methods in solving stiff differential equations. We also analyse the order of convergence and stability property for the general form. Moreover, we give its corresponding vector form via introducing a new definition of vector product and quotient in another article. The convergence and stability of the vector form are considered as well.     

Review of explicit Falkner methods and its modifications for solving special second-order I.V.P.s

The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented.     

Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems

For the general multidimensional perturbed oscillators y^″+Ky=f(y,y^′) with K∈Rd×d, the order conditions for the ARKN methods are presented by X. Wu et al. [Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250–2257]. These methods integrate exactly the multidimensional unperturbed oscillators and are highly efficient when the perturbing function is small. In this paper, we are concerned with the analysis of the concrete multidimensional ARKN methods based on the order conditions for the multidimensional ARKN methods. We extent the matrix K to a general case, in which K is not required to be symmetric. Numerical experiments demonstrate that the novel multidimensional ARKN methods presented in this paper are more efficient compared with some well-know RKN methods in the scientific literature.     

Spline methods for solving system of second-order boundary-value problems

In this paper, we use cubic polynomial splines to derive some consistency relations which are then used to develop a numerical method for computing smooth approximations to the solution and its derivatives for a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

Generalized newton multi-step iterative methods GMNp,m for solving system of nonlinear equations

A generalization of the Newton multi-step iterative method is presented, in the form of distinct families of methods depending on proper parameters. The proposed generalization of the Newton multi-step consists of two parts, namely the base method and the multi-step part. The multi-step part requires a single evaluation of function per step. During the multi-step phase, we have to solve systems of linear equations whose coefficient matrix is the Jacobian evaluated at the initial guess. The direct inversion of the Jacobian it is an expensive operation, and hence, for moderately large systems, the lower-upper triangular factorization (LU) is a reasonable choice. Once we have the LU factors of the Jacobian, starting from the base method, we only solve systems of lower and upper triangular matrices that are in fact computationally economical. The developed families involve unknown parameters, and we are interested in setting them with the goal of maximizing the convergence order of the global method. Few families are investigated in some detail. The validity and numerical accuracy of the solution of the system of nonlinear equations are presented via numerical simulations, also involving examples coming from standard approximations of ordinary differential and partial differential nonlinear equations. The obtained results show the efficiency of constructed iterative methods, under the assumption of smoothness of the nonlinear function.     

The effect of global extrapolation on the phase-lag of symmetric methods for solving periodic initial value problems

The effects are discussed of two-grid global extrapolation procedures on the phase-lags of convergent numerical methods for solving periodic initial value problems.

The procedure is tested on two methods applied to two problems from the literature, one nonlinear the other linear, and the effects of the extrapolation are examined by comparing corresponding zeros of the waves generated by the theoretical and computed solutions.

Extensions to three- and four-grid extrapolation procedures are outlined in an Appendix.     

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.     

Higher order dispersive and dissipative hybrid method for the numerical solution of oscillatory problems

In this paper, an improved explicit two-step hybrid method with fifth algebraic order is derived. The new method possesses dispersion of order 10 and dissipation of order seven, which is first of its kind in the literature. Numerical experiment reveals the superiority of the new method for solving oscillatory or periodic problems over several methods of the same algebraic order.     

One-step explicit methods for the numerical integration of perturbed oscillators

In this paper, we present a class of one-step explicit zero-dissipative nonlinear methods for the numerical integration of perturbed oscillators, which have second algebraic order and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods.