Author ORCID Identifier
0009-0006-8150-157X
Document Type
Dissertation
Date of Award
8-31-2024
Degree Name
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Department
Mathematical Sciences
First Advisor
David Shirokoff
Second Advisor
Cristina Frederick
Third Advisor
Travis Askham
Fourth Advisor
Simone Marras
Fifth Advisor
David I. Ketcheson
Abstract
Numerical stability is a critical property for a time-integration scheme. In the context of Runge-Kutta methods applied to stiff differential equations, A-stability is one of the most basic and practically important notions of stability. Dating back to the work of Dahlquist, it has been known that A-stability is equivalent to the Runge-Kutta stability function satisfying a particular convex feasibility problem. Specifically, up to a transformation, the stability function lies in the convex cone of positive functions. In recent years, sum-of-squares optimization and semidefinite programming have become valuable tools in developing rigorous certificates of stability in dynamical systems. Therefore, it is natural to employ these convex optimization tools for the purpose of rigorously certifying A- and A(α)-stability in Runge-Kutta methods.
Two distinct convex feasibility problems defined by linear matrix inequalities are introduced. The first approach employs sum-of-squares programming applied to the Runge-Kutta E-polynomial, making it applicable to both A- and A(α)-stability. The second approach refines the algebraic conditions for A-stability, as developed by Cooper, Scherer, Turke, and Wendler (CSTW), to incorporate the Runge-Kutta order conditions. The theoretical enhancement of the algebraic conditions facilitates the practical application of the refined conditions for certifying A-stability within a computational framework.
Additionally, a new theoretical perspective is provided, relating the algebraic conditions for A-stability to continued fraction approximations of the exponential. This perspective involves the introduction of a new transform defined in a recently established class of polynomials orthogonal with respect to a linear functional.
The E-polynomial and CSTW methodologies are utilized to obtain rigorous stability certificates for several implicit Runge-Kutta schemes proposed in the literature. Specific attention is given to certifying the implicit Runge-Kutta schemes utilized in the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS).
Recommended Citation
Juhl, Austin, "Certifying stability in runge-kutta schemes: Algebraic conditions and semidefinite programming" (2024). Dissertations. 1775.
https://digitalcommons.njit.edu/dissertations/1775
Included in
Numerical Analysis and Computation Commons, Numerical Analysis and Scientific Computing Commons
