PDE Realization and Structure-Aware Solvers for a Compact Two-Stage Fourth-Order IMEX Method

This paper develops a PDE realization and solver framework for a compact two-stage fourth-order two-derivative implicit–explicit time discretization for stiff split problems. The key difficulty is a consistency–cost coupling: full-field temporal differentiation is required to preserve the mixed explicit–implicit interactions of the time integrator, but the same interactions widen the implicit stage operators and can make each solve substantially more expensive. We show that a second-order ADER/Cauchy–Kowalevski local evolution is sufficient for the fourth-order outer composition, establish smooth reconstruction consistency and a fully discrete error balance, and derive the leading Lie-bracket defect produced by self differentiation of the split fields. An inexact-stage analysis gives asymmetric midpoint and endpoint residual tolerances that preserve fourth-order accuracy. For the widened stages, the complete mixed action is retained matrix-free while only dominant stiff physics is approximated in the inverse. This yields quadratic and shifted preconditioners, Fourier and multilevel realizations, a semilinear reaction–diffusion reduction, and source-local elimination for relaxation systems. Exact quadratic cancellation, a diffusion-dominated Fourier estimate, and an $\epsilon \mathrm{/}\mathrm{\Delta }x$ bound for Jin–Xin relaxation explain the main solver mechanisms. Numerical ablations verify the consistency and tolerance results, while a two-dimensional Brusselator study on two grids shows favorable error-versus-wall-time behavior over a useful accuracy range. A classical stiff-front benchmark also identifies the separate spatial shock–source limitation. The results support a regime-dependent efficiency claim rather than a universal speedup.