Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time

TL;DR

This work unifies Projective Integration (PI) methods for stiff ODEs with large spectral gaps by casting them as Runge-Kutta (RK) schemes with extended Butcher tableaux. This RK reformulation enables rigorous analysis of consistency, accuracy, and stability, and provides a framework for spatial and time adaptivity via partitioned and embedded RK methods. The authors derive time-adaptive PI schemes from embedded RK methods and step-size variation, and they introduce stable on-the-fly error estimators by adopting RK structures, while demonstrating instability in the original Gear-based approach. Numerical tests on a two-scale model confirm the theoretical results, showing improved stability regions and competitive convergence, especially for embedded and partitioned-PI variants. The results pave the way for more robust, adaptive PI schemes applicable to multi-scale problems in science and engineering.

Abstract

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge-Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge-Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge-Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge-Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Figures (12)

• Figure 1: Space Adaptive Forward Euler scheme (SAFE) with small time step $\delta t$ in stiff region (left) and large time step $\Delta t$ in non-stiff region (right). Values of red cells at the boundary of the two domains need to be reconstructed, see KoellermeierAPI.
• Figure 2: Space Adaptive Projective Forward Euler scheme (SAPFE) with $K+1$ inner small time steps $\delta t$ in stiff region (left) and large time step $\Delta t$ in non-stiff region (right). Values of red cells at the boundary of the two domains need to be reconstructed, see KoellermeierAPI.
• Figure 3: Space Adaptive Projective Projective Forward Euler scheme (SAPPFE) with $K+1=3$ inner small time steps $\delta t_L$ in stiff region (left) and$K+1=3$ inner small time steps $\delta t_R > \delta t_L$ in semi stiff region (right). Values of red cells at both sides of the boundary of the two domains need to be reconstructed, see KoellermeierAPI.
• Figure 4: Projective outer step size variation (POSV) for Projective Forward Euler scheme with $K=2$. Top: standard PFE with outer step size $\Delta t$. Bottom: PFE with outer step size $\Delta t/2$.
• Figure 5: Last projective inner step size variation (PISV) for Projective Forward Euler scheme. Top: standard PFE with inner step size $\delta t$. Bottom: PFE with last inner step using two steps of size $\delta t/2$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Theorem 3
• proof
• Remark 2
• Remark 3
• Remark 4