Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization
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DOI：
10.1016/j.cma.2016.09.035
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发表时间：
2017-01-01
影响因子：
7.2
通讯作者：
Bonilla, Jesus
中科院分区：
文献类型：
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作者：
Badia, Santiago;Bonilla, Jesus
在这项工作中，我们提出了一个非线性稳定技术的标量守恒律隐式时间步进。该方法依赖于人工扩散方法，基于图形拉普拉斯算子。它是非线性的，因为它依赖于冲击检测器。此外，所得方法是线性保持的。使用相同的冲击检测器来逐渐集中质量矩阵。由此产生的方法是LED，正性保持，也满足一个全球性的。Lipschitz连续性也得到了证明。然而，由此产生的计划是高度非线性的，导致非常差的非线性收敛速度。我们提出了一个光滑的版本的计划，这导致两次可微的非线性稳定计划。它允许人们直接使用牛顿法并获得二次收敛。在数值实验中，考虑了二维稳态和瞬态线性输运，以及瞬态Burgers方程。利用牛顿法和光滑格式，我们可以减少10到20倍的迭代次数的安德森加速与原来的非光滑格式。在任何情况下，这些性质只对收敛解成立，而对迭代不成立。在这个意义上，我们还提出了投影非线性求解器的概念，其中投影步骤在每次非线性迭代结束时执行到容许解的Ph空间上。容许解的空间是满足期望的单调性质（极大值原理或正性）的空间。(C)© 2016 Elsevier B.V.版权所有。
In this work, we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping. The method relies on an artificial diffusion method, based on a graph-Laplacian operator. It is nonlinear, since it depends on a shock detector. Further, the resulting method is linearity preserving. The same shock detector is used to gradually lump the mass matrix. The resulting method is LED, positivity preserving, and also satisfies a global DMP. Lipschitz continuity has also been proved. However, the resulting scheme is highly nonlinear, leading to very poor nonlinear convergence rates. We propose a smooth version of the scheme, which leads to twice differentiable nonlinear stabilization schemes. It allows one to straightforwardly use Newton's method and obtain quadratic convergence. In the numerical experiments, steady and transient linear transport, and transient Burgers' equation have been considered in 2D. Using the Newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of Anderson acceleration with the original non-smooth scheme. In any case, these properties are only true for the converged solution, but not for iterates. In this sense, we have also proposed the concept of projected nonlinear solvers, where a projection step is performed at the end of every nonlinear iterations onto a Ph space of admissible solutions. The space of admissible solutions is the one that satisfies the desired monotonic properties (maximum principle or positivity). (C) 2016 Elsevier B.V. All rights reserved.