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Boundary conditions for kinetic theory‐based models II: A linearized moment system

This work is concerned with boundary conditions (BCs) for a linearized hyperbolic moment system of the Boltzmann equation in one dimension. We show that even if the usual relaxation stability conditions and the Kreiss condition hold, there exists an exponentially increasing solution to the initial‐boundary‐value problem (IBVP) of the moment system. To clarify this problem, we check the generalized Kreiss condition (GKC) for hyperbolic relaxation systems. With the GKC, the stability of the moment system is proved by using an energy estimate together with the Laplace transformation. Moreover, under the GKC, we derive the reduced boundary conditions for the corresponding equilibrium system. Numerical results verify the convergence of the solution of the moment system to that of the equilibrium system with the derived BCs in the relaxation limit. Our analysis indicates that special attention should be paid when imposing boundary conditions for moment systems and the GKC should be respected to ensure the zero relaxation limit of the IBVP.