Multistep Collocation Research Articles

Boundary value technique for initial value problems based on Adams-type second derivative methods

In this article, we propose a family of second derivative Adams-type methods (SDAMs) of order up to 2k + 2 (k is the step number) for initial value problems. The methods are constructed through a continuous approximation of the SDAM which is obtained by multistep collocation. The continuous approximation is used to obtain initial value methods, which are simultaneously applied to generate all approximations on the entire interval. The order and the linear stability properties of the methods are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the literature.

Multistep collocation methods for Volterra Integral Equations

We introduce multistep collocation methods for the numerical integration of Volterra Integral Equations, which depend on the numerical solution in a fixed number of previous time steps. We describe the constructive technique, analyze the order of the resulting methods and their linear stability properties. Numerical experiments confirm the theoretical expectations.

Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis

Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation, is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria for the choice of this filter are investigated numerically.

Numerical experiments with a multistep Radau method

This paper describes an implementation of multistep collocation methods, which are applicable to stiff differential problems, singular perturbation problems, and D.A.E.s of index 1 and 2. These methods generalize one-step implicit Runge-Kutta methods as well as multistep one-stage BDF methods. We give numerical comparisons of our code with two representative codes for these methods, RADAU5 and LSODE.

The stability function for multistep collocation methods

C-polynomials for rational approximation to the exponential function was introduced by NOrsett [7] to study stability properties of one-step methods. For one-step collocation methods theC-polynomial has a very simple form. In this paper we studyC-polynomials for multistep collocation methods and obtain results that generalize those in the one-step case, and provide a way to analyze linear stability of such methods.

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Norsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.

Superconvergence for multistep collocation

One-step collocation methods are known to be a subclass of implicit Runge-Kutta methods. Further, one-leg methods are special multistep one-point collocation methods. In this paper we extend both of these collocation ideas to multistep collocation methods with k previous meshpoints and m collocation points. By construction, the order is at least m + k − 1 m + k - 1 . However, by choosing the collocation points in the right way, order 2 m + k − 1 2m + k - 1 is obtained as the maximum. There are ( m + k − 1 k − 1 ) \left ( {\begin {array}{*{20}{c}} {m + k - 1} \\ {k - 1} \\ \end {array} } \right ) sets of such "multistep Gaussian" collocation points.