Vol. 7, No. 2, 2012

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
Optimal stability polynomials for numerical integration of initial value problems

David I. Ketcheson and Aron J. Ahmadia

Vol. 7 (2012), No. 2, 247–271
Abstract

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.

Keywords
absolute stability, initial value problems, Runge–Kutta methods
Mathematical Subject Classification 2010
Primary: 65L06, 65M20
Secondary: 90C26
Milestones
Received: 12 July 2012
Revised: 23 November 2012
Accepted: 28 November 2012
Published: 8 January 2013
Authors
David I. Ketcheson
Division of Mathematical and Computer Sciences and Engineering
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Aron J. Ahmadia
Division of Mathematical and Computer Sciences and Engineering
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia