An L-Stable Implicit Two-Stage Fourth-Order Temporal Discretization Scheme for Lax-Wendroff-Type Solvers Applied to Stiff Problems

TL;DR

This paper introduces an implicit two-stage fourth-order temporal discretization scheme for Lax-Wendroff-type solvers, enabling larger stable time steps and improved accuracy for stiff problems.

Contribution

It develops a novel implicit TSFO scheme with L-stability, faster convergence, and potential for enhanced flow-field capturing, surpassing classical methods.

Findings

Achieves fourth-order accuracy in two stages.

Allows larger stable time steps than classical methods.

Reduces convergence errors by an order of magnitude.

Abstract

The explicit two-stage fourth-order (TSFO) temporal-spatial coupling method is efficient and compact but suffers severe time-step restrictions for stiff problems with multiple scales. To address Professor Jiequan Li's call for an implicit extension, this paper first constructs an implicit TSFO time discretization scheme using the method of undetermined coefficients and Taylor expansion. Second, using a model equation and the maximum modulus principle, sufficient conditions for L-stability are derived. Third, a Newton iteration accelerates convergence. Numerical experiments on classical stiff benchmarks show that the proposed implicit scheme achieves fourth-order temporal accuracy in two stages. Compared to the classical fourth-order implicit Runge-Kutta method, it allows larger stable time steps and reduces convergence errors by an order of magnitude. More importantly, this implicit scheme can be extended to construct an implicit TSFO temporal-spatial coupling method that captures flow-field correlations and handles strong discontinuities, fundamentally contrasting with method-of-lines approaches. Additionally, it unlocks Lax-Wendroff-type solvers to naturally and synchronously embed both stiff source terms and flow transport into time derivatives, thereby avoiding operator-splitting errors.