Turing-region preservation in matrix-oriented splitting methods for reaction-diffusion systems
We develop matrix-oriented formulations of first-order splitting integrators for two-species reaction-diffusion systems on 2D domains, using the Gierer-Meinhardt system as a benchmark for discrete Turing instability. Exploiting the differential matrix equation associated with tensor-product spatial discretizations, we obtain the families IE-S and EX-S, where the diffusive flow is treated by implicit Euler or exactly, respectively, and the reaction substep is approximated by explicit, symplectic, adjoint-symplectic, Poisson, and explicit-variant local maps. Starting from the continuous diffusion-driven instability threshold, expressed through the modal relation $J_\mu = J^* - \mu D$, we derive fully discrete modal amplification matrices and their Jury conditions. These conditions separate the continuous Turing mechanism, carried by the first Jury condition, from discrete effects carried by the second. Specializing the analysis to the Gierer-Meinhardt model, we exhibit two opposite pathologies. First, in a continuous Turing-stable regime, IMEX may generate a stable spurious pattern through a violation of the second Jury condition. Second, in a real continuous Turing regime, the adjoint-symplectic family may be spuriously stable and suppress the pattern that IMEX correctly detects. For IMEX, whose first Jury condition reproduces exactly the sign of the continuous Turing polynomial, we give an explicit time-step condition guaranteeing preservation of the continuous Turing region, and show that controlling the second Jury condition is needed only in the Turing-stable regime. These examples show that each integrator induces its own discrete Turing region, which should be compared with the continuous one before interpreting numerical patterns; we frame this requirement as preserving a qualitative property of the continuous problem, in the spirit of structure-preserving numerical integration.