Zhixin Huo
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.