Obrechkoff versus super-implicit methods for the solution of first- and second-order initial value problems

AbstractThis paper discusses the numerical solution of first-order initial value problems and a special class of second-order ones (those not containing first derivative). Two classes of methods are discussed, super-implicit and Obrechkoff. We will show equivalence of super-implicit and Obrechkoff schemes. The advantage of Obrechkoff methods is that they are high-order one-step methods and thus will not require additional starting values. On the other hand, they will require higher derivatives of the right-hand side. In case the right-hand side is complex, we may prefer super-implicit methods. The disadvantage of super-implicit methods is that they, in general, have a larger error constant. To get the same error constant we require one or more extra future values. We can use these extra values to increase the order of the method instead of decreasing the error constant. One numerical example shows that the super-implicit methods are more accurate than the Obrechkoff schemes of the same order

Numerical methods for ordinary differential equations with applications to partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis develops a number of algorithms for the numerical solution of ordinary differential equations with applications to partial differential equations. A general introduction is given; the existence of a unique solution for first order initial value problems and well known methods for analysing stability are described. A family of one-step methods is developed for first order ordinary differential equations. The methods are extrapolated and analysed for use in PECE mode and their theoretical properties, computer implementation and numerical behaviour, are discussed. Lo-stable methods are developed for second order parabolic partial differential equations 1n one space dimension; second and third order accuracy is achieved by a splitting technique in two space dimensions. A number of two-time level difference schemes are developed for first order hyperbolic partial differential equations and the schemes are analysed for Ao-stability and Lo-stability. The schemes are seen to have the advantage that the oscillations which are present with Crank-Nicolson type schemes, do not arise. A family of two-step methods 1S developed for second order periodic initial value problems. The methods are analysed, their error constants and periodicity intervals are calculated. A family of numerical methods is developed for the solution of fourth order parabolic partial differential equations with constant coefficients and variable coefficients and their stability analyses are discussed. The algorithms developed are tested on a variety of problems from the literature.British Governmen

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school

The stability of laminar incompressible boundary layers in the presence of compliant boundaries

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1964.Microfiche copy available in Barker.Vita.Includes bibliographical references (leaves 118-122).by Richard E. Kaplan.Sc.D

P-stable high-order super-implicit and Obrechkoff methods for periodic initial value problems

The article of record as published may be found at http://dx.doi.org/10.1016/j.camwa.2005.11.041This paper discusses the numerical solution of periodic initial value problems. Two classes of methods are discussed, superimplicit and Obrechkoff. The advantage of Obrechkoff methods is that they are high-order one-step methods and thus will not require additional starting values. On the other hand they will require higher derivatives of the right-hand side. In cases when the right-hand side is very complex, we may prefer super-implicit methods. We develop a super-implicit P-stable method of order 12 and Obrechkoff method of order 18

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal