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A new nonlinear L-stable scheme with constant and adaptive step-size strategy

Arain Sadia, Qureshi Sania, Shaikh Asif Ali

Journal of Applied Mathematics and Computational Mechanics

2021

Vol. 20, nr 4

7--18

The present study proposes a new explicit nonlinear scheme that solves stiff and nonlinear initial value problems in ordinary differential equations. One of the promising features of this scheme is its fourth-order convergence with strong stability having an unbounded region. A modern approach for relative stability growth analysis is also presented under order stars conditions. The scheme is also good in dealing with singular and stiff type of models. Comparing numerical experiments using various errors, including maximum absolute global error over the integration interval, absolute error at the endpoint, average error, norm of errors, and the CPU times (seconds), shows better performance. An adaptive step-size approach seems to increase the performance of the proposed scheme. The numerical simulations assure us of L -stability, consistency, order, and rapid convergence.

Local error structures and order conditions in terms of Lie elements for exponential splitting schemes

Auzinger W., Herfort W.

Opuscula Mathematica

2014

Vol. 34, no. 2

243--256

We discuss the structure of the local error of exponential operator splitting methods. In particular, it is shown that the leading error term is a Lie element, i.e., a linear combination of higher-degree commutators of the given operators. This structural assertion can be used to formulate a simple algorithm for the automatic generation of a minimal set of polynomial equations representing the order conditions, for the general case as well as in symmetric settings.