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

Multi-parameter exponentially fitted, P-stable Obrechkoff methods

We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for second order differential equations. An earlier result for single-parameter exponential fitting is re-examined and extended to multi-parameter, multi-order exponential fitting

Boundary value problems and dichotomic stability

Since the conditioning of a boundary value problem (BVP) is closely related to the existence of a dichotomic fundamental solution (i.e., where one set of modes is increasing and a complementary set is decreasing), it is important to have discretization methods that conserve this dichotomy property. The conditions this imposes on such a method are investigated in this paper. They are worked out in more detail for scalar second-order equations (the central difference scheme), and for linear first-order systems as well; for the latter type both one-step methods (including collocation) and multistep methods (those that may be used in multiple shooting) are examin

A Functionally-Fitted Block Numerov Method for Solving Second-Order Initial-Value Problems with Oscillatory Solutions

[EN] A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the general second-order ordinary differential equation, y′′=f(x,y,y′), it is a fourth order convergent method for the special second-order ordinary differential equation, y′′=f(x,y). Comparison with other methods in the literature, even of higher order, shows the good performance of the proposed method.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Finite element multistep multideriavative schemes for parabolic equations

The linear, homogeneous, parabolic equation is solved by applying finite element discretizations in space and A0 —stable, linear multistep, multiderivative (L.M.S.D.) methods in time. Such schemes are unconditionally stable. An error analysis establishes an optimal bound in the L2 —norm. Methods typifying the class of L.M.S.D. schemes are derived and their implementation examined

Projected explicit and implicit Taylor series methods for DAEs

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.Peer Reviewe

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