Rate Constant Matrix Contraction Method for Stiff Master Equations with Detailed Balance

TL;DR

This paper provides a mathematical foundation and improved implementation of the rate constant matrix contraction (RCMC) method for solving large, stiff master equations with detailed balance, validated through numerical experiments.

Contribution

It reformulates the RCMC method in matrix terms, offers a theoretical error analysis, and discusses efficient, stable implementation strategies.

Findings

Validated the method's efficiency and stability on synthetic models

Demonstrated accuracy in real kinetic models

Provided a rigorous mathematical foundation for RCMC

Abstract

This paper considers master equations for Markovian kinetic schemes that possess the detailed balance property. Chemical kinetics, as a prime example, often yields large-scale, highly stiff equations. Based on chemical intuitions, Sumiya et al. (2015) presented the rate constant matrix contraction (RCMC) method that computes approximate solutions to such intractable systems. This paper aims to establish a mathematical foundation for the RCMC method. We present a reformulated RCMC method in terms of matrix computation, deriving the method from several natural requirements. We then perform a theoretical error analysis based on eigendecomposition and discuss implementation details caring about computational efficiency and numerical stability. Through numerical experiments on synthetic and real kinetic models, we validate the efficiency, numerical stability, and accuracy of the presented method.