Work

On the stability of Runge–Kutta methods for arbitrarily large systems of ODEs

Year: 2024

Type: article

Abstract: We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based on the numerical range of the underlying large matrices involved in such systems, and demonstrate its application with concrete examples of RK stability for hyperbolic methods of lines.

Source: Communications on Pure and Applied Mathematics

DOI: 10.1002/cpa.22238

Author: Eitan Tadmor

Institution: University of Maryland, College Park

Language: English

Topic: Numerical methods for differential equations

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,980

Funders: Fondation Sciences Mathématiques de Paris, Office of Naval Research