The choice on an efficient direct integration procedure for linear structural dynamic equations of motion is discussed. An explicit symmetric multistep method is presented. By combining these two ideas, we are able to construct P-stable methods of arbitrary even order. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Fr die Schrdinger-Gleichung wird eine neue exponentiell angepate Zweischrittmethode hergeleitet und angewendet. The new formulae are derived by exponential fitting, and they represent a generalization of the usual Gauss—Laguerre formulae.
In this work we deal with exponentially fitted methods for the numerical solution of second order ordinary differential equations, whose solutions are known to show a prominent exponential behaviour depending on the value of an unknown parameter to be suitably determined. The method is useful only when a good estimate of the frequency of the problem is known in advance. Some numerical tests are presented. Exponential Fitting is a procedure for an efficient numerical approach of functions consisting of weighted sums of exponential, trigonometric or hyperbolic functions with slowly varying weight functions. We illustrate the theoretical results using some numerical experiences. In this paper those definitions are modified so as to provide a basis for linear stability analysis of exponential-fitting methods for the special class of ordinary differential equations of second order in which the first derivative does not appear explicitly. A new class of quadrature formulae for the computation of integrals over unbounded intervals with oscillating integrand is illustrated.
Only few of them, however, take advantage of special properties of the solution that may be known in advance. The authors studied the field for many years and contributed to it. These methods, which depend on a parameter, can be constructed following a six-step flow chart of Ixaru and Vanden Berghe. Various modified Störmer—Cowell methods have been proposed to overcome this deficiency, but they all require a priori knowledge of the frequency. They will, however, not be considered in this paper.
A direct quadrature method for the solution of Volterra integral equations with periodic solution is proposed. Exponential is also of or involving an exponent or exponents. The basic characteristic of these methods is that their phase lag vanishes at a predefined frequency. New quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope. The asymptotic order is discussed, and an algorithm for determining weights and nodes for a general number N of nodes is provided, resulting an improvement of the existing quadrature formulae.
The book can also be used as a textbook for graduate students. Such methods are needed in various branches of natural science, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. For the produced methods we investigate their errors and stability. In this paper a new explicit Numerov-type method is introduced. It is also shown that the stability functions of the thus obtained methods can be expressed as Padé approximants of the exponential function with a complex argument.
The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Numerical results obtained for well-known test problems show the efficiency of the new method. A new two-step exponentially-fitted formula is derived and applied to the Schrdinger equation. Other definition of exponential is raised to the power of e, the base of natural logarithms exp. The coefficients depend on the frequency of oscillation, in order to improve the accuracy of the solution. Numerical experiments illustrate the proposed strategy.
The error analysis indicates that each version will be very different when big values of the energy are involved. Mathematics and Its Applications, vol 568. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver. Some related research: Runge-Kutta-Nystroem methods. This paper deals with the construction of a fourth-order symplectic exponentially fitted modified Gauss method. The stability and phase properties are examined. Some numerical experiments are presented for comparison with other existing methods.
Due to the multistage nature of these methods, the proposed technique takes into account the contribution to the error arising from the computation of the internal stages. The first definition of exponential in the dictionary is of, containing, or involving one or more numbers or quantities raised to an exponent, esp ex. This method is of higher algebraic order and is fitted both exponentially and trigonometrically. It is shown that if the integrand is analytic, then in the absence of stationary points, the method is rapidly convergent. The new method is found to significantly more accurate than the standard methods, for large values of the energy. The work is mainly concerned with two-stepmethods but extensions tomethods of larger step-number are also considered.
In this book, a series of aspects is covered, ranging from the theory of the procedure up to direct applications and sometimes including ready to use programs. Operations on the functions described above like numerical differentiation, quadrature, interpolation or solving ordinary differential equations whose solution is of this type, are of real interest nowadays in many phenomena as oscillations, vibrations, rotations, or wave propagation. It is found that the new methods are P-stable. Exponential fitted algorithms for initial value and boundary value methods and for the calculation of quadrature rules are introduced. Department of Applied Mathematics and Computer Science University of Gent Gent Belgium About this chapter Cite this chapter as: Ixaru L.