Macroscale modelling of multilayer diffusion: Using volume averaging to correct the boundary conditions

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• We study the boundary conditions (BCs) in a macroscale model of layered diffusion. • A set of corrected macroscale BCs are derived using the method of volume averaging. • For example: a Dirichlet BC at the microscale leads to a Robin BC at the macroscale. • Resulting macroscale field more accurately captures averaged microscale field. • Resulting reconstructed field is in excellent agreement with true microscale field. This paper investigates the form of the boundary conditions (BCs) used in macroscale models of PDEs with coefficients that vary over a small length-scale (microscale). Specifically, we focus on the one-dimensional multilayer diffusion problem, a simple prototype problem where an analytical solution is available. For a given microscale BC (e.g., Dirichlet, Neumann, Robin, etc.) we derive a corrected macroscale BC using the method of volume averaging. For example, our analysis confirms that a Robin BC should be applied on the macroscale if a Dirichlet BC is specified on the microscale. The macroscale field computed using the corrected BCs more accurately captures the averaged microscale field and leads to a reconstructed microscale field that is in excellent agreement with the true microscale field. While the analysis and results are presented for one-dimensional multilayer diffusion only, the methodology can be extended to and has implications on a broader class of problems.