通过参数化改进牛顿法性能： Richards方程的案例
Improving Newton's method performance by parametrization: the case of Richards equation
AI总结：
本文通过参数化压力与饱和度的关系图，改进 Richards方程中牛顿法的稳定性，并证明了数值方案的正确性及局部二次收敛性。
摘要：
由退化抛物方程离散得到的非线性系统可能难以求解，尤其是使用牛顿法时。本文将一种策略应用于Richards方程，即定义新的主未知数以稳定牛顿法，通过参数化压力与饱和度的图来实现。然后利用单调有限体积方案对修正后的Richards方程进行离散化。我们证明了数值方案的正确性。然后在适当的非退化条件下，证明牛顿法局部二次收敛。最后，我们提供了本方法有效性的数值证据。
英文摘要：
The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newtonâs method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach.
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