Propagation of internal errors in explicit Runge--Kutta methods and internal stability of SSP and extrapolation methods

TL;DR

This paper analyzes how internal errors propagate within explicit Runge--Kutta methods, revealing that implementation choices significantly affect stability and providing bounds on error amplification for various classes of methods.

Contribution

It offers a general analysis of internal error propagation in Runge--Kutta methods, highlighting the impact of implementation details and providing bounds for error amplification.

Findings

Internal error propagation can be catastrophic in Runge--Kutta methods.

Implementation details significantly influence internal stability polynomials.

Bounds on error amplification constants are provided for methods with many stages.

Abstract

In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.