Optimal stability polynomials for numerical integration of initial value problems

TL;DR

This paper develops a new convex optimization-based algorithm to find optimal stability polynomials for numerical integration, maximizing stable step size for initial value problems with known spectra.

Contribution

It introduces a fast, accurate, and robust algorithm for computing optimal stability polynomials, with proven convergence in specific cases and demonstrated effectiveness beyond those cases.

Findings

Algorithm achieves global convergence for certain spectra

Effectively computes stability polynomials for various spectra

Improves stability step size in numerical integration

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.