Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

TURNING POINT PROBLEMS FOR THE THIRD ORDER DIFFERENTIAL EQUATIONS EXHIBITING RESONANCE

ＴＵＲＮＩＮＧＰＯＩＮＴＰＲＯＢＬＥＭＳＦＯＲＴＨＥＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＥＸＨＩＢＩＴＩＮＧＲＥＳＯＮＡＮＣＥＸＩＡＯＣＨＥＮＧＹＯＵ（肖成猷）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＥａｓｔＣｈｉ...     

Unconditionally stable methods for second order differential equations

We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2, ...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them.     

Construction of two-step Runge-Kutta methods of high order for ordinary differential equations

The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane. This revised version was published online in August 2006 with corrections to the Cover Date.     

Supplement: Efficient weak second order stochastic Runge-Kutta methods for non-commutative Stratonovich stochastic differential equations

This paper gives a modification of a class of stochastic Runge-Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.     

Numerical treatment of retarded differential–algebraic equations by collocation methods

This paper investigates retarded differential–algebraic equations of index zero to two with state-dependent delay. The theory needed to understand the numerical approach and analyze the numerical treatment by collocation methods is developed. Different strategies for tracking the jump discontinuities are considered and numerical examples are presented to support the convergence results. This revised version was published online in August 2006 with corrections to the Cover Date.     

Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter . The coefficients in the expansion of the global error in powers of are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.     

Error of rosenbrock methods for stiff problems studied via differential algebraic equations

This paper studies Rosenbrock methods when they are applied to stiff differential equations containing a small stiffness parameter. The basic ideas and techniques are the same as those developed for Runge-Kutta methods in an earlier paper of the authors. The results obtained here are essentially those obtained for Diagonally Implicit Runge-Kutta methods.     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

High order methods for the numerical integration of ordinary differential equations

Summary High order implicit integration formulae with a large region of absolute stability are developed for the approximate numerical integration of both stiff and non-stiff systems of ordinary differential equations. The algorithms derived behave essentially like one step methods and are demonstrated by direct application to certain particular examples.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.     

Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.     

Diagonally implicit general linear methods for ordinary differential equations

We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, p=q orq+1,s=r+1=q, p=q orq+1, ands=r=q, p=q+1. Examples of such methods are constructed with stability function matching theA-acceptable generalized Padé approximations to the exponential function.The work of Z. Jackiewicz was partially supported by the National Science Foundation under grant NSF DMS-9208048.     

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

Summary The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.     

Digital total variation filtering as postprocessing for Chebyshev pseudospectral methods for conservation laws

Digital total variation filtering is analyzed as a fast, robust, post-processing method for accelerating the convergence of pseudospectral approximations that have been contaminated by Gibbs oscillations. The method, which originated in image processing, can be combined with spectral filters to quickly post-process large data sets with sharp resolution of discontinuities and with exponential accuracy away from the discontinuities.     

Numerical methods for the eigenvalue determination of second-order ordinary differential equations

An accurate method for the numerical solution of the eigenvalue problem of second-order ordinary differential equation using the shooting method is presented. The method has three steps. Firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method. Secondly the initial-value problem is solved using new, highly accurate formulas of the linear multistep method. Thirdly the eigenvalue is properly corrected at the matching point. The efficiency of the proposed methods is demonstrated by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Pöschl–Teller potential in quantum mechanics.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.