Optimal Stability Polynomials for Numerical Integration of Initial Value Problems
Optimal Stability Polynomials for Numerical Integration of Initial Value Problems
By David Ketcheson, and Aron Jamil Ahmadia
arXiv.org (2012)
Abstract Paper

David Ketcheson

King Abdullah University of Science and Technology

United States

Coder Page  

Aron Jamil Ahmadia

Columbia University

United States

Coder Page  

This code reproduces the figures 3a, 3b, 4a, 4b, 6a, 6b and 7b of the article “Optimal stability polynomials for numerical integration of initial value problems” (David I. Ketcheson and Aron J. Ahmadia, 2012). The user can fix the number of stages (s) and the order p, and then retrieve (1) the scaled size of real axis interval inclusion for optimized methods Hopt/s^2 (Table 1, page 12), (2) the scaled size of imaginary axis inclusion for optimized methods Hopt/s (Table 2, page 13), (3) the relative size of largest disk that can be included in the stability region scaled by the number of stages (Figure 5, page 15) and (4) the optimal effective step size (Figure 7a, page 16). Please note that updated code is available at http://numerics.kaust.edu.sa/RK-opt/
Created
July 30, 2012
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Matlab R2008
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December 01, 2012
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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.
Ketcheson, D., and A. J. Ahmadia, "Optimal Stability Polynomials for Numerical Integration of Initial Value Problems", arXiv.org.
Coders:
  • David Ketcheson

    King Abdullah University of Science and Technology

    United States

  • Aron Jamil Ahmadia

    Columbia University

    United States

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