Abstract The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving near degenerate nonlinear systems. To solve them, we need robust and efficient linearization schemes. Apart from Newton's method, which fails to converge for large time steps and small mesh sizes, recently the globally convergent first order L-scheme was developed, which approximates the derivative by a global upper bound improving Picard's scheme. In this paper, we extend this method to build a family of robust efficient globally convergent first-order linearization schemes by using a sequence of real functions L n , n ≥ 0 . We solve the space variable by a piecewise linear Galerkin finite element method. The time discretization is based on the Forward-Backward Euler method in the term of hydraulic conductivity K, whereas the remaining terms of the equation dealt with the Backward Euler approximation. We get five schemes improving the convergence of L-schemes and are getting closer to a second-order convergent method. We prove the time iteration convergence in the H 1 ( Ω ) norm, and we test the schemes on numerical benchmarks to compare them with the L-scheme and Newton's scheme. Among the new schemes, four of them are globally convergent and one scheme is a hybrid case with Newton's scheme. Our results show that among the new schemes, Modified Generalized Linear Scheme (MGLS) obtains the best convergence rates (up to four times that of the L-scheme) using fewer iteration steps and less overall computing time. Finally, our hybrid scheme is robust since it converges when Newton's scheme does not and uses similar computing time, which makes it a good alternative for locally convergent second-order schemes. More methods can be developed using this framework with other L n sequences and spatial discretizations.

中文翻译：

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用于求解理查兹方程的一系列新的全局收敛线性化方案

摘要 Richards 方程模拟地下水含水层和石油储层模拟中通过多孔介质的水流。这是一个退化抛物线微分方程，其中没有解析解是已知的。大多数数值方法使用支持大时间步长的隐式时间方案，但涉及接近退化的非线性系统。为了解决这些问题，我们需要稳健有效的线性化方案。除了牛顿法在大时间步长和小网格尺寸下无法收敛外，最近还开发了全局收敛一阶 L 方案，它通过改进皮卡德方案的全局上限来近似导数。在本文中，我们扩展了这种方法，通过使用一系列实函数 L n 来构建一系列稳健高效的全局收敛一阶线性化方案，n≥0。我们通过分段线性伽辽金有限元方法求解空间变量。时间离散化基于水力传导率 K 项中的 Forward-Backward Euler 方法，而方程的其余项处理 Backward Euler 近似。我们得到了五种改进 L-schemes 收敛的方案，并且越来越接近二阶收敛方法。我们证明了 H 1 ( Ω ) 范数下的时间迭代收敛性，并在数值基准上测试了这些方案，以将它们与 L 方案和牛顿方案进行比较。在新方案中，有四个方案是全局收敛的，一个方案是与牛顿方案的混合案例。我们的结果表明，在新方案中，修改后的广义线性方案 (MGLS) 使用更少的迭代步骤和更少的总计算时间获得最佳收敛速度（高达 L 方案的四倍）。最后，我们的混合方案是稳健的，因为它在牛顿方案不收敛时收敛，并且使用相似的计算时间，这使其成为局部收敛二阶方案的一个很好的替代方案。可以使用该框架与其他 L n 序列和空间离散化来开发更多方法。