We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate reduced-order models. Proper orthogonal decomposition is used to construct a reduced-basis (RB) space which provides a rapidly convergent approximation of the parametric solution manifold. The Galerkin projection is employed to reduce the dimensionality of the problem by projecting the weak formulation of the governing PDEs onto the RB space. A major challenge in model reduction for nonlinear PDEs is the efficient treatment of nonlinear terms, which we address by unifying the implementation of several hyperreduction methods. We introduce a first-order empirical interpolation method to approximate the nonlinear terms and recover the computational efficiency. We demonstrate the effectiveness of our methodology through its application to the Allen-Cahn equation, which models phase separation processes, and the Buckley-Leverett equation, which describes two-phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed approach.

翻译：本文提出一种针对含参数依赖的时间依赖非线性偏微分方程（PDEs）实时求解的模型降阶方法。该方法融合多种技术要素，以构建高效且精确的降阶模型。通过本征正交分解构建降维基（RB）空间，该空间能够快速收敛地逼近参数化解流形。采用Galerkin投影法，将控制偏微分方程的弱形式投影至RB空间，从而实现问题维度的约简。非线性偏微分方程模型降阶面临的主要挑战在于非线性项的高效处理，本文通过统一实现多种超降阶方法以应对此问题。我们引入一阶经验插值方法来逼近非线性项并恢复计算效率。通过将该方法应用于模拟相分离过程的Allen-Cahn方程，以及描述多孔介质中两相流体流动的Buckley-Leverett方程，我们验证了所提方法的有效性。数值结果突显了该方案在精度、效率及稳定性方面的优越性能。