Symmetry-Preserving Strang Splitting for Stiff Relaxation Systems

Operator splitting methods are widely used for efficient simulation of multiscale systems with disparate time scales. Although classical schemes such as Strang splitting are formally second order, they often suffer accuracy degradation in stiff relaxation systems when the fast dynamics are temporally under-resolved. In this work, we revisit Strang splitting in stiff regimes from a geometric perspective, leveraging the slow-manifold structure induced by rapid relaxation. Our analysis reveals that in resolved regimes, the substep composition exhibits a hidden symmetry that enables error cancellation and thereby underpins Strang’s second-order accuracy; in under-resolved stiff regimes, the relaxation substep induces a stiff-limit attraction toward quasi-equilibrium, which disrupts the stage-wise cancellation mechanism and leads to order reduction. Motivated by this mechanism, we introduce a symmetry-preserving modification that retains the Strang composition and replaces only the stiff relaxation substep with a time-reversible implicit-midpoint update. The resulting scheme preserves the cancellation mechanism and achieves uniform second-order accuracy for the slow dynamics, even in strongly under-resolved stiff regimes. Numerical experiments on a linear damped oscillator, a nonlinear reaction--diffusion system, and the Bhatnagar--Gross--Krook model corroborate the analysis: classical Strang splittings exhibit order reduction when under-resolved, whereas the proposed method maintains robust second-order accuracy for the slow dynamics and captures the correct first-order stiff-limit behavior of the fast dynamics over a wider range of time steps, allowing larger steps for a prescribed accuracy.