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
Division of Mathematical and
Computer Sciences and Engineering
King Abdullah University of Science and Technology
(KAUST)
Thuwal 23955-6900
Saudi Arabia
Division of Mathematical and
Computer Sciences and Engineering
King Abdullah University of Science and Technology
(KAUST)
Thuwal 23955-6900
Saudi Arabia