A Two-Stage Fourth-Order Implicit Scheme for Stiff problems
Zhixin Huo
TL;DR
This work introduces a two-stage, fourth-order implicit scheme (Implicit TSFO) designed for stiff problems, leveraging temporal derivatives within a spatiotemporal (GRP/ADER) framework to achieve high accuracy in two stages. A stability analysis yields an A-stability condition with a specific parameter range and an optimal choice, supported by Newton-iteration-based implementation. Numerical tests on linear stiff systems, Robertson kinetics, ozone decomposition, and the Van der Pol oscillator demonstrate superior stability and reduced error compared to conventional implicit Runge-Kutta methods, confirming the method's effectiveness for multi-scale problems. The approach promises practical impact for solving stiff ODEs and PDEs with strong source terms in fluid dynamics and reactive flows, especially when spatiotemporal coupling is essential.
Abstract
This paper presents a novel two-stage fourth-order implicit scheme designed to overcome the limitations of existing spatiotemporal coupling two-stage fourth-order methods, which have thus far been confined to explicit frameworks. In such frameworks, computational efficiency is severely constrained by the CFL condition when addressing stiff problems, and they fundamentally conflicts with spatiotemporal coupled methods whose core advantage lies in embedding stiff source terms. By fully leveraging temporal derivatives of physical variables within a rigorously derived compact mathematical framework, the scheme achieves fourth-order temporal accuracy in only two stages. Furthermore, a sufficient condition for A-stability is established through systematic theoretical and numerical investigation, and a Newton iterative procedure is provided to accelerate convergence. Extensive numerical experiments on classical stiff problems confirm the method's effectiveness and competitiveness.