Superconvergent explicit two-step peer methods

Abstract

We consider explicit two-step peer methods for the solution of nonstiff differential systems. By an additional condition a subclass of optimally zero-stable methods is identified that is superconvergent of order p = s + 1, where s is the number of stages. The new condition allows us to reduce the number of coefficients in a numerical search for good methods. We present methods with 4-7 stages which are tested in FORTRAN90 and compared with DOPRI5 and DOP853. The results confirm the high potential of the new class of methods.

Parallel start for explicit parallel two-step peer methods

“…Such methods have been constructed in [6,9] with the additional aims of good stability, small error constants and moderate coefficient magnitudes. An additional topic was an order increase to s + 1 by a certain superconvergence property [10]. However, in all implementations so far, the estimate for the initial stepsize and the required starting values Y 0i have been computed by Runge-Kutta methods.…”

“…The multiplication with the matrix A m can be expressed by simple operations for high order AB(s + 1). (10) and let p be the interpolating polynomial (14). Then the interpolating polynomial r of Y m = M m (z)Y m−1 after one step of the method (2) applied to y = λy is given by…”

“…The index m indicates that (some of) these coefficients may depend on the time step. This class of peer methods has been introduced recently in its non-parallel form using derivatives h m j<i r ij f (t mj , Y mj ) from the actual time step in [9] and discussed further in [6,10]. (3) closely resembles earlier versions of General Linear Methods as in the first edition of Butcher's textbook [1].…”

Exponentially fitted fifth-order two-step peer explicit methods

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Abstract.The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced by R. Weiner et al [6,7,11,12,13] for solving different types of problems either in sequential or parallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces with dimension six.

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On the derivation of explicit two-step peer methods

“…Weiner and Schmitt and co-workers have provided for some special A-matrix methods of type (3) of different orders [10], [17], [18] that are competitive with the standard integrators in use.…”