Using Lobatto nodes, one-step methods of order six and eight have been obtained for the second-order differential equation y″ = f(x, y), y(x0) = y0, y′(x0) = y′0. The methods are shown to be P-stable. If , then at each integration step a system of dimension 3s, 4s, respectively, has to be solved. The numerical results, for two problems, obtained by using these methods are given in the end. 相似文献
A five-stage fifth-order singly diagonally implicit Runge–Kutta–Nyström method for the integration of second order differential equations possessing an oscillatory solution, is presented in this article. This method is P-stable, which is recommended for problems with a theoretical solution consisting of a periodic part of moderate frequency with a high frequency oscillation with small amplitude superimposed. It also attains an order which is one higher than existing methods of this type. Numerical comparisons with existing methods of this type show its clear advantage. 相似文献
In this work we introduce a new family of 12-step linear multistep methods for the integration of the Schr?dinger equation.
The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both
the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the
integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error
analysis and numerical applications.
T. E. Simos is a highly cited researcher, active member of the European Academy of Sciences and Arts.
Corresponding member of the European Academy of Sciences, corresponding member of European Academy of Arts, Sciences and Humanities. 相似文献
Following the ideas of Ananthakrishnaiah we develop a family of P-stable Obrechkoff methods of arbitrary even order. The coefficients
of these methods follow from a recursive algorithm. 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. A numerical example is given to
illustrate the performance of the methods. 相似文献
In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution
of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and
its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications. 相似文献
A generator of new embedded P-stable methods of order 2n+2, where n is the number of layers used by the embedded methods, for the approximate numerical integration of the one-dimensional Schrödinger equation is developed in this paper. These new methods are called embedded methods because of a simple natural error control mechanism. Numerical results obtained for one-dimensional differential equations of the Schrödinger type show the validity of the developed theory. 相似文献
We consider the construction of P-stable exponentially-fitted symmetric two-step Obrechkoff methods for solving second order
differential equations related to an initial value problem. Our approach is based on two ideas: for the exponential fitting,
we follow the ideas of Ixaru and Vanden Berghe; for the P-stability we introduce exponentially-fitted Padé approximants to
the exponential function. By combining these two ideas, we are able to construct P-stable methods of arbitrary (even) order. 相似文献
In this paper, we investigate the numerical performance of a family of
P-stable two-step Maruyama schemes in mean-square sense for stochastic differential
equations with time delay proposed in [8, 10] for a certain
class of nonlinear stochastic delay differential equations
with multiplicative white noises.
We also test the convergence of one of the schemes for
a time-delayed Burgers' equation with an additive white noise. Numerical results show that
this family of two-step Maruyama methods exhibit similar stability for nonlinear equations
as that for linear equations. 相似文献