In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method solves a reduced order linear system in order to compute the Newton step. This reduced system is derived using a projection matrix, obtained using proper orthogonal decomposition, which is updated in each time step of the numerical method. We demonstrate that the method can be used together with Euler's implicit method to simulate CO$_2$ injection into an oil reservoir, and we compare with using Newton's method. The Newton-like method achieves a speedup of between 39% and 84% for systems with between 4,800 and 52,800 state variables.

翻译：本文提出一种基于模型降阶技术的类牛顿法，可用于逼近常微分方程解的隐式数值方法。在每次迭代中，该类牛顿法通过求解一个降阶线性系统来计算牛顿步。该降阶系统利用通过本征正交分解获得的投影矩阵推导得到，该投影矩阵在数值方法的每个时间步长中更新。我们证明该方法可与欧拉隐式法结合，用于模拟CO₂注入油藏的过程，并与牛顿法进行对比。对于具有4800至52800个状态变量的系统，该类牛顿法可实现39%至84%的速度提升。