Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge--Kutta methods

TL;DR

This paper develops a new family of embedded SSP Runge-Kutta methods of orders 2 to 4, enabling adaptive step-size control for hyperbolic PDEs while preserving stability and accuracy.

Contribution

It introduces a novel construction of embedded pairs for SSP explicit Runge-Kutta methods, enhancing adaptive step-size control without losing stability.

Findings

Effective error estimation for adaptive step sizing.

Maintains strong stability preserving properties.

Balances accuracy and computational efficiency.

Abstract

We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.