README.md

2025H

Codex/ChatGPT (July 2026)

Summary

Explicit two-stage fourth-order (TSFO) methods are compact but become inefficient for stiff multiscale problems because stability constrains the time step. This paper constructs an implicit TSFO temporal discretization using Taylor expansions and undetermined coefficients. A model-equation analysis with the maximum-modulus principle yields sufficient conditions for $L$-stability, so strongly damped modes decay as required for stiff integration. Newton iteration supplies a practical nonlinear solution procedure. Classical stiff benchmarks verify fourth-order temporal convergence using only two stages. In the reported comparisons, the method accepts larger stable time steps and produces errors roughly an order of magnitude below a classical fourth-order implicit Runge--Kutta method. The proposed time discretization is designed as a foundation for implicit Lax--Wendroff-type solvers, where temporal derivatives can incorporate transport and stiff sources together and thereby avoid operator-splitting error. That fully coupled temporal-spatial extension, including strong-discontinuity calculations, is presented as subsequent work rather than established here.

Contributions

1. Constructed an implicit two-stage method with fourth-order temporal accuracy.
2. Derived sufficient $L$-stability conditions from a model-equation analysis and the maximum-modulus principle.
3. Formulated a Newton iteration for solving the method's implicit stage equations.
4. Verified fourth-order convergence and stiff stability on classical benchmark problems.
5. Established a temporal framework for unsplit Lax--Wendroff-type treatment of flow transport and stiff source terms.