Open Access
2012 Optimal stability polynomials for numerical integration of initial value problems
David Ketcheson, Aron Ahmadia
Commun. Appl. Math. Comput. Sci. 7(2): 247-271 (2012). DOI: 10.2140/camcos.2012.7.247


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.


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David Ketcheson. Aron Ahmadia. "Optimal stability polynomials for numerical integration of initial value problems." Commun. Appl. Math. Comput. Sci. 7 (2) 247 - 271, 2012.


Received: 12 July 2012; Revised: 23 November 2012; Accepted: 28 November 2012; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1259.65114
MathSciNet: MR3020216
Digital Object Identifier: 10.2140/camcos.2012.7.247

Primary: 65L06 , 65M20
Secondary: 90C26

Keywords: absolute stability , initial value problems , Runge–Kutta methods

Rights: Copyright © 2012 Mathematical Sciences Publishers

Vol.7 • No. 2 • 2012
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