High‐order hierarchical A‐ and L‐stable integration methods

Abstract

SUMMARYNumerical integration of stiff first-order systems of differential equations is considered. It is shown how a Co-continuous polynomial time discretization, in conjunction with a weighted residual method, can be used to derive methods corresponding to the diagonal and first subdiagonal Pade approximants. In this manner A-stable schemes of order 2k and L-stable schemes of order 2k -1 are obtained, at least for k -5 4.The methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower-order methods that have the same stability type. We show how this feature may be utilized to perform partial refinements in the integration process, and how an error estimation by an embedding approach naturally follows. An adaptive algorithm based on the error estimator is suggested. Some numerical experiments that illustrate the ideas are included.

Algorithms by design with illustrations to solid and structural mechanics/dynamics

“…The subtle issues may not be readily apparent for all these Type 3 k q methods unless one carefully evaluates the totality of the developments in the context of the algorithmic measures and algorithmic complexity just as we have explicitly demonstrated in the case of LMS methods. Also, algorithmic structures can be designed in hierarchical fashion [47,56] to improve the efficiency; however, the type of algorithmic relationships of these formulations in contrast to those of the standard architectures needs to be clearly understood. As a result, the unified theory and the design spaces and measures provides overall guidelines for both the novelty of the contributions and/or assessing the quality of time operators.…”

“…And, the different classes of algorithms that may result from amongst any one of a variety of approaches and the design spaces and the associated algorithmic structures of the Type 1, Type 2, and Type 3 classification are illustrated in Figures 2-4. For example, as shown in Reference [1], the LMS methods [5], the serial sub-stepping method of the Tarnow-Simo algorithm [44], the parallel sub-stepping algorithm [45], the Runge-Kutta type sub-stepping methods [46] that are customarily implemented with various stages in the solution process, and the generalized weighted residual methods [47][48][49] all belong to the classification of the Type 3 design space. The spectral equivalence relation that is established between the Type 3 k q classification and the corresponding Type 2 k (p, q) algorithms is presented here for the first time to provide an improved understanding and to enable establishing the relations amongst the Type 3 k q classification theoretically.…”

A new unified theory underlying time dependent linear first‐order systems: a prelude to algorithms by design

“…A simple extension to the previous approach originally due to Möller [32] employing a time weighted residual approach for establishing a mapping relation between a particular polynomial rational form Type 2 k (p, q) algorithm constructed by using instead a linear combination of the diagonal and the first sub-diagonal Padé approximation for the exact amplification matrix and the corresponding Type 3 k q representation as described in References [23,33] which belongs to the present framework is described. Consider…”

Complex-time-step methods for transient analysis

“…Hence, the evaluations can be computed in parallel with no message transfer between two evaluation processes. It is interesting to note that the derived higher order methods do not require nested equation systems to be solved (as in the implicit Runge}Kutta methods [1,2,5], the discontinuous Galerkin method [13}15], or the Petrov}Galerkin method [16], etc.) or matrix multiplication (as in the case of Hermitian discertization [17] or the direct application of PadeH approximations [18,19]).…”