Reduced-order transport models are widely used in computational physics to describe diffusion, exchange, relaxation, and redistribution processes when full spatially resolved models are unnecessary, unavailable, or computationally expensive. This paper develops a sign-structured matrix framework for a two-compartment diffusion-relaxation system, the smallest nontrivial model combining environmental relaxation with bidirectional inter-compartment exchange. Starting from a physically interpretable mass-balance formulation, the system is written as a linear state-space model. A diagonal signature transformation converts the governing matrix into a matrix with nonpositive entries and strictly positive determinant for all positive physical parameters. In two dimensions, this is equivalent to the property that the determinant is positive while all proper minors are nonpositive. The same parameter regime is shown to imply no singularity, asymptotic stability of the original diffusion dynamics, and physically meaningful relaxation toward equilibrium. The exact matrix-exponential solution is used to interpret the spectral decay modes, while a forward Euler discretization is analyzed to connect continuous-time stability with time-step-dependent numerical stability and first-order global convergence. Computational experiments confirm decay toward equilibrium and slope-one convergence of the discretization error. The results suggest that sign-structured matrix analysis can serve as a compact framework for linking local dissipative coupling, global solvability, spectral stability, and numerical reliability in reduced-order diffusive transport models.

Reduced-order transport models are widely used in computational physics to describe diffusion, exchange, relaxation, and redistribution processes when full spatially resolved models are unnecessary, unavailable, or computationally expensive. This paper develops a sign-structured matrix framework for a two-compartment diffusion-relaxation system, the smallest nontrivial model combining environmental relaxation with bidirectional inter-compartment exchange. Starting from a physically interpretable mass-balance formulation, the system is written as a linear state-space model. A diagonal signature transformation converts the governing matrix into a matrix with nonpositive entries and strictly positive determinant for all positive physical parameters. In two dimensions, this is equivalent to the property that the determinant is positive while all proper minors are nonpositive. The same parameter regime is shown to imply no singularity, asymptotic stability of the original diffusion dynamics, and physically meaningful relaxation toward equilibrium. The exact matrix-exponential solution is used to interpret the spectral decay modes, while a forward Euler discretization is analyzed to connect continuous-time stability with time-step-dependent numerical stability and first-order global convergence. Computational experiments confirm decay toward equilibrium and slope-one convergence of the discretization error. The results suggest that sign-structured matrix analysis can serve as a compact framework for linking local dissipative coupling, global solvability, spectral stability, and numerical reliability in reduced-order diffusive transport models.