First‐Order Empirical Interpolation Method for Real‐Time Solution of Parametric Time‐Dependent Nonlinear PDEs

Abstract

We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations(PDEs) with parametric dependencies. A major challenge in constructing efficient and accurate reduced-order models for nonlin-ear PDEs is the efficient treatment of nonlinear terms. We address this by unifying the implementation of hyperreduction methodsto deal with nonlinear terms. Furthermore, we introduce a first-order empirical interpolation method (EIM) to provide an effi-cient approximation of the nonlinear terms in time-dependent PDEs. We demonstrate the effectiveness of our approach on theAllen–Cahn equation, which models phase separation, and the Buckley–Leverett equation, which describes two-phase fluid flowin porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed method compared with boththe Galerkin–Newton approach and hyper-reduced models using the standard EIM.