# Implementing state-space methods for multizone contaminant transport

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### Abstract

The “well-mixed zone” approximation is a useful model for simulating contaminant transport in buildings. Multizone software tools such as CONTAM [1] and COMIS [2] use time-marching numerical methods to solve the resulting ordinary differential equations. By contrast, the state-space approach solves the same equations analytically [3]. A direct analytical solution, using the matrix exponential, is computationally attractive for certain applications, for example, when the airflows do not change for relatively long periods. However, for large systems, even the matrix exponential requires numerical estimation. This paper evaluates two methods for finding the matrix exponential: eigenvalue decomposition, and the Padé algorithm. In addition, it considers a variation optimised for sparse matrices, and compares against a reference backward Euler time-marching scheme.

The state-space solutions can run several orders of magnitude faster than the reference method, with more significant speedups for a greater number of zones. This makes them especially valuable for applications where rapid calculation of concentration and exposure under constant air flow conditions are needed, such as real-time forecasting or monitoring of indoor contaminants. For most models, all three methods have low errors (magnitude of median fractional bias <3·10−5, normalised mean square error <3·10−7, and scaled absolute error <4·10−4). However, for the largest model considered (1701 zones) eigenvalue decomposition showed a dramatic increase in error.