Perturbative analytical framework for thermal wave diffusion in non-linear building envelopes

Model Predictive Control (MPC) in building energy management requires transient thermal models balancing thermodynamic accuracy with computational efficiency. Standard spatial discretization triggers state-space inflation, paralyzing real-time solvers, while Transfer Matrix Methods (TMM) suffer from high-frequency numerical overflow and assume material homogeneity. This paper introduces a novel frequency-domain framework based on the continuous spatial Riccati equation. A recursive admittance mapping strictly bounds exponential growth, preventing numerical instability. The primary contribution of this work is extending this linear propagator via regular perturbation theory to analytically resolve spatial property gradients ( ⁡( )) and non-linear 4 radiative boundaries as equivalent harmonic source terms. This meshless approach eliminates spatial truncation errors. It corrects peak heating load deviations of 21.9% in wetted media and mitigates artificial nocturnal cooling fluxes of 12.0 W/m2. Preserving ⁡( ) spatial complexity, the framework avoids state-space inflation, ensuring high-speed execution for multi-week MPC optimization.