Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge--Kutta methods
This work provides a practical tool for researchers solving hyperbolic PDEs, enabling adaptive time-stepping while preserving nonlinear stability, though it is an incremental improvement over existing SSP methods.
The authors construct embedded pairs for optimal SSP explicit Runge-Kutta methods of orders 2-4, enabling adaptive step-size control for hyperbolic PDEs. Numerical experiments demonstrate effectiveness in balancing precision, work, accuracy, and stability.
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.