An L-Stable Sequential Two-Stage Fourth-Order Method with ADER Trajectory Derivatives for Stiff Transport--Relaxation Systems

Abstract

A fully implicit two-stage fourth-order two-derivative time discretization was introduced previously as a temporal method. This paper closes that sequential integrator for stiff transport--relaxation equations by pairing a conservative finite-volume residual ${\mathcal{L}}_h$ with its discrete trajectory derivative ${\mathcal{G}}_h^{\mathrm{tr}}=D{\mathcal{L}}_h{\mathcal{L}}_h$. An ADER/Cauchy--Kowalevski predictor provides interface states and physical time derivatives; differentiating the same numerical flux and taking shared face differences yields a conservative approximation ${\tilde{\mathcal{G}}}_h$. For linear constant-coefficient balance laws, ${\tilde{\mathcal{G}}}_h={\mathcal{G}}_h^{\mathrm{tr}}={\mathcal{L}}_h^2$ exactly, although the derivative operator is assembled independently rather than by squaring the residual matrix. For nonlinear discretizations, the fourth-order temporal theory applies to ${\mathcal{G}}_h^{\mathrm{tr}}$, while a trajectory-closure consistency estimate controls the ADER approximation. The two unknown stage vectors are solved successively through two $N$-unknown systems. The completed step is fourth order and L-stable; the parameter $C_q=5/183$ cancels the leading inverse-power term and changes the deep-stiff amplification from $O(|z{|}^{-1})$ to $O(|z{|}^{-2})$. For fixed compatible spatial spaces, a slow--fast decomposition proves a full-step asymptotic-preserving operator limit with an $O(\delta )$ estimate and gives a preparation-dependent uniform-accuracy classification. Linear finite-volume, nonlinear relaxation, one- and two-dimensional damping, diffusion-limit, and modal experiments verify the corresponding closure, accuracy, stability, and singular-limit claims within their stated scopes.