Off-step Points Research Articles

Strong-stability-preserving 3-stage Hermite–Birkhoff time-discretization methods

Strong-stability-preserving (SSP) time-discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws. A collection of SSP explicit 3-stage Hermite–Birkhoff methods of orders 3 to 7 with nonnegative coefficients are constructed as k-step analogues of third-order Runge–Kutta methods, incorporating a function evaluation at two off-step points. Generally, these new methods have larger effective CFL coefficients than the hybrid methods of Huang with the same step number k. They have larger maximum scaled step sizes than hybrid methods on Burgers' equations.

Sextic spline solution of variable coefficient fourth-order parabolic equations

We report new three-level implicit methods of O(k 2+h 4) and O(k 4+h 4) for the numerical solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficients. We use sextic spline in space and finite difference in time directions. Sextic spline function relations are derived by using off-step points. The linear stability of the presented method is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existence methods shows the superiority of our presented scheme.

Quintic Spline Methods for the Solution of Singularly Perturbed Boundary-Value Problems

We develop a class of methods for the numerical solution of singularly perturbed two-point boundary–value problems. Using non-polynomial quintic spline in grid points we can obtain the fourth–order method only. But we develop the quintic spline in off-step points to raise the order of accuracy. Based on such spline, the purposed methods are fourth, sixth, and eighth–order accurate. These methods are applicable to problems both in singular and non-singular cases. The convergence analysis of the fourth–order method is stablished. Numerical results are given to illustrate the efficiency of our methods.

On the hybrid method with three off-step points for initial value problems

A continuous representation of a hybrid method with three ‘off-step’ points is developed via interpolation and collocation procedures, and used to obtain initial value methods (IVMs) for solving initial value problems. The IVMs are assembled into a single block matrix equation which is convergent and A-stable. We note that accuracy is improved by applying the IVMs as simultaneous numerical integrators on sub-intervals which do not overlap. The method is compared with the existing ones in the literature and its superiority over the fourth-order Runge–Kutta method is established numerically.

Hybrid BDF methods for the numerical solutions of ordinary differential equations

In this article, we have presented the details of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solutions of ordinary differential equations (ODEs). In these hybrid BDF, one additional stage point (or off-step point) has been used in the first derivative of the solution to improve the absolute stability regions. Stability domains of our presented methods have been obtained showing that all these new methods, we say HBDF, of order p, p = 2,4,..., 12, are A(α)-stable whereas they have wide stability regions comparing with those of some known methods like BDF, extended BDF (EBDF), modified EBDF (MEBDF), adaptive EBDF (A-EBDF), and second derivtive Enright methods. Numerical results are also given for five test problems.

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic equations (DAEs) of arbitrarily high fixed index. A scheme to generate first-order derivatives at off-step points is combined with Pryce scheme which generates high order derivatives at step points. The stepsize is controlled by a local error estimator. HBT(10)9DAE uses only the first four derivatives of y instead of the first 10 required by Taylor’s series method T10DAE of order 10. Dormand–Prince’s DP(8,7)13M for ODEs is extended to DP(8,7)DAE for DAEs. HBT(10)9DAE wins over DP(8,7)DAE on several test problems on the basis of CPU time as a function of relative error at the end of the interval of integration. An index-5 problem is equally well solved by HBT(10)9DAE and T10DAE. On this problem, the error in the solution by DP(8,7)DAE increases as time increases.

Stiffly stable continuous extension of second derivative linear multi-step methods with an off-step point for initial value problems in ordinary differential equations.

In this paper, we introduce a continuous extension of second derivative linear multi-step methods with a hybrid point for the numerical solution of initial valued stiff ordinary differential equations. The continuous extension is based on the Gear's fixed step size backward differential methods [7]. The intervals of absolute stability of methods of step number k&#88047 are determined using the root locus plot. Numerical results of the methods solving a non-linearly stiff initial value problem in ordinary differential equations are compared with that from the state-of-the-art ordinary differential equations code of MATLAB discussed in Higham et al [9]. JONAMP Vol. 11 2007: pp. 175-190

Strong stability preserving hybrid methods

This paper is concerned with the strong stability preserving (SSP) time discretizations for semi-discrete systems, obtained from applying the method of lines to time-dependent partial differential equations. We focus on the construction of explicit hybrid methods with nonnegative coefficients, which are a class of multistep methods incorporating a function evaluation at an off-step point. A series of new SSP methods are found. Among them, the low order methods are more efficient than some well known SSP Runge–Kutta or linear multistep methods. In particular, we present some fifth to seventh order methods with nonnegative coefficients, which have healthy CFL coefficients. Finally, some numerical experiments on the Burgers equation are given.

Nordsieck Methods with an Off-Step Point

Hybrid methods, incorporating one or more off-step points, are difficult to implement in a variable stepsize situation using the standard representation of input and output data in each step. However, instead of representing this data in terms of solution values and derivative values at a sequence of step points, it is possible to reformulate the method so that it operates on a Nordsieck vector. This has the consequence of reducing stepsize adjustments to nothing more than rescaling the components of the Nordsieck vector. This paper shows how to derive methods in both formulations and considers some implementation details. It is also possible to derive a new type of hybrid method using the Norsieck representation as the starting point and this is also discussed in the paper. The new method is found to have comparable accuracy for corresponding work expended as for standard methods.

Parallel block pc methods with rkn-type correctors and adams-type predictors*

This paper describes the construction of block predictor - corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor - corrector method not only at step point, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-accurate predictions using Adams-type formulas. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictions using last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor - corrector methods are comparable with those of a conventional predictor - corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor -corrector methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the best parallel and sequential methods from the literature.

RKN-type parallel block PC methods with Lagrange-type predictors

This paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nyström correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nyström corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor-corrector methods to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature.

Variable-order, variable-step methods for second-order initial-value problems

We develop a variable-order, variable-step algorithm for solving second-order initial-value problems y″ = f(t, y) , y(0) , y′(0) given. This algorithm employs a family of implicit two-step methods, in which the differential equation is evaluated at a number of off-step points. We have derived the order conditions for this family of methods in previously published work and we have also shown that the remaining free parameters of the method may be chosen to given P-stability. Here, the emphasis is on the combination of second-, fourth-, sixth- and eighth-order methods from this family in a variable-order, variable-step algorithm. The aim is to choose particular P-stable methods from the family in such a way that a saving can be made in the amount of computational effort involved when they are combined together.

Generalized Hermite multistep methods of high order with a variable stepsize

Generalized Hermite multistep methods for initial-value problems in ordinary differential equations are reviewed. Zero-stability of the correctors with variable stepsize is investigated. A special predictor-corrector scheme for multistep methods with offstep points is presented. A method of order 11 has been developed and is illustrated with numerical examples.

Parallel block predictor-corrector methods of Runge-Kutta type

In this paper, we construct block predictor-corrector methods using Runge-Kutta correctors. Our approach consists of applying the predictor-corrector method not only at step points, but also at off-step points (block points), so that, in each step, a whole block of approximations to the exact solution is computed. In the next step, these approximations are used to obtain a high-order predictor formula by Lagrange or Hermite interpolation. By choosing the abscissas of the off-step points narrowly spaced, a much more accurately predicted value is obtained than by predictor formulas based on proceding step point values. Since the approximations at the off-step points to be computed in each step can be obtained in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta correctors, the predictor-corrector iteration scheme itself is also highly parallel. Application of these block predictor-corrector methods based on Lagrange-Gauss pairs to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 2 to 11 when compared with the best sequential methods.

Two high-order multistep methods with offstep points applied with a variable stepsize

In this paper we derive new generalized multistep methods with variable stepsize for initial-value problems in ordinary differential equations. Some numerical examples are given.

A block 6(4) Runge-Kutta formula for nonstiff initial value problems

A new selection is made for an efficient two-step block Runge-Kutta formula of order 6. The new formula is developed using some of the efficiency criteria recently investigated by Shampine, and as a result, a block formula with much improved performance is obtained. An important property of this new formula is that there is a “natural” interpolating polynomial available. This can be used to compute approximate solution values at off-step points without the need to compute any additional function evaluations. The quality of this interpolant is examined, and it is shown to have certain desirable properties. The performance of the new block Runge-Kutta formula is evaluated using the DETEST test set and is shown to be more efficient than certain other standard Runge-Kutta formulas for this particular test set.

Block embedded explicit runge-kutta methods

Embedded explicit Runge—Kutta formulae are amongst the most popular methods currently in use for the approximate numerical integration of nonstiff systems of ordinary differential equations. In particular the fourth-order methods RKF45 of Shampine and Watts and the Runge—Kutta—Merson code of the NAG Library have been particularly popular for some time. In this paper we consider the derivation of a block embedded explicit Runge—Kutta (BERK) formula of order 4. BERK formulae have all the characteristics of standard explicit Runge—Kutta formulae except that they are no longer single step in nature in the sense that a pth order BERK formula produces pth order approximations to the solution at several step points instead of at one point only. The results of some fairly extensive numerical testing are presented and these indicate that the new formula is very competitive, both in terms of efficiency and reliability, with the codes R—K Merson of NAG and RKF45. Also considered in detail are the problems of computing solutions at “off-step” points and of efficiently computing low accuracy solutions. In particular we show that BERK formulae allow intermediate solutions at “off-step” points to be computed with relatively little additional computational effort. This makes BERK formulae particularly attractive for problems where output is requested at many off-step points since it is for this class of problems that conventional Runge—Kutta formulae can become very inefficient. Also considered is the question of computing low accuracy solutions. It is shown that, as is to be expected, such problems are solved more efficiently using low order BERK formulae. In particular, BERK formulae of orders 1 and 3 are derived and their performance is compared with that of the fourth-order BERK formula on a large set of test problems.

An implicit off-step point method for the integration of stiff differential equations

Recently England introducedk-step formulas of orderk+2 which use one off-step point for the integration of stiff differential equations. These formulas are slightly modified, theorems on the error constants are given and numerical results are presented.

Higher order difference methods for second order two-point boundary-value problems

Using methods of numerical integration, difference schemes of sixth order with off-step points have been obtained and applied to the second order differential equation, with and without mixed boundary conditions. Numerical results obtained by these methods have been compared with those obtained by using h4-extrapolation of the Numerov method. It is found that the sixth-order method based upon 4-point Lobatto quadrature with approximation 2 is more economical and accurate from the computational viewpoint than the existing sixth-order methods.

High Order Numerical Differentiation Type Formulas with an Off-step Point for Stiff Ordinary Differential Equations

High Order Numerical Differentiation Type Formulas with an Off-step Point for Stiff Ordinary Differential Equations