Acceleration of nonlinear solvers for natural convection problems
Acceleration of nonlinear solvers for natural convection problems
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DOI：
10.1515/jnma-2020-0067
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发表时间：
2020-04
影响因子：
3
通讯作者：
Sara N. Pollock;L. Rebholz;Mengying Xiao
中科院分区：
文献类型：
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作者：
Sara N. Pollock;L. Rebholz;Mengying Xiao
摘要 本文使用应用于 Picard 迭代的安德森加速，为非等温流的稳定 Boussinesq 模型开发了一种高效且稳健的求解技术。在分析与非线性迭代相关的定点算子以证明某些稳定性和规律性属性成立之后，我们应用了作者最近构建的安德森加速理论，该理论产生了 Boussinesq 系统的安德森加速皮卡德迭代的收敛结果。结果表明，残差中的首项因优化问题中的增益而得到改善，但代价是额外的高阶项，当残差很大时，这些高阶项可能会很显着。我们进行数值测试来说明该理论，并表明安德森深度的两阶段选择可能是有利的。我们还考虑将安德森加速应用于 Boussinesq 方程的牛顿迭代，并观察到，即使使用标准线搜索，加速也可以使牛顿迭代收敛到比没有加速时明显更高的瑞利数。
Abstract This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.