Algebraic Stabilization of Explicit Numerical Integration for Extremely Stiff Reaction Networks

TL;DR

This paper demonstrates that explicit numerical methods, stabilized by algebraic solutions, can efficiently solve extremely stiff reaction networks, challenging the dominance of implicit methods especially in large-scale simulations.

Contribution

It introduces algebraic stabilization techniques for explicit methods applicable to stiff reaction networks, including near and far from equilibrium scenarios, with validation on astrophysical thermonuclear networks.

Findings

Explicit stabilized methods handle the stiffest networks effectively.

These methods achieve accuracy comparable to implicit methods.

Explicit methods offer faster computation and better scalability.

Abstract

In contrast to the prevailing view in the literature, it is shown that even extremely stiff sets of ordinary differential equations may be solved efficiently by explicit methods if limiting algebraic solutions are used to stabilize the numerical integration. The stabilizing algebra differs essentially for systems well-removed from equilibrium and those near equilibrium. Explicit asymptotic and quasi-steady-state methods that are appropriate when the system is only weakly equilibrated are examined first. These methods are then extended to the case of close approach to equilibrium through a new implementation of partial equilibrium approximations. Using stringent tests with astrophysical thermonuclear networks, evidence is provided that these methods can deal with the stiffest networks, even in the approach to equilibrium, with accuracy and integration timestepping comparable to that of implicit methods. Because explicit methods can execute a timestep faster and scale more favorably with network size than implicit algorithms, our results suggest that algebraically-stabilized explicit methods might enable integration of larger reaction networks coupled to fluid dynamics than has been feasible previously for a variety of disciplines.