Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of partial differential equations with time-varying and nonlinear boundary conditions. Control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial. However, numerical simulations of high-order discretized partial differential equations are often too computationally expensive for real-time and many-query analysis. For this reason, model complexity reduction is essential. In this paper, it is shown that the classical reduced basis method is unable to incorporate time-varying and nonlinear boundary conditions. To address this issue, it is shown that, by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input-related term, the effects of the boundary conditions are accurately described in the reduced-order model. The results are compared with the classical reduced basis method. Unlike the classical method, the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries. Moreover, a new approximation of the bound and a new estimate for the error induced by model reduction are introduced. The effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided. The proposed approximate error bound gives a finite bound of the actual error, unlike existing error bounds that grow exponentially over time. Finally, the proposed error estimate is more accurate than existing error estimates. •Control inputs usually act the boundaries of the system.•Correct implementation of boundary effects at the reduced order level is important.•Correct incorporation of boundary conditions is illustrated in this paper.•An approximate error bound is proposed to quantify the error due to the reduction.•A sharp error estimate based on the approximate error bound is introduced.