Numerical reduced basis methods are instrumental to solve parameter dependent partial differential equations problems in case of many queries. Bifurcation and instability problems have these characteristics as different solutions emerge by varying a bifurcation parameter. Rayleigh–Bénard convection is an instability problem with multiple steady solutions and bifurcations by varying the Rayleigh number. In this paper the eigenvalue problem of the corresponding linear stability analysis has been solved with this method. The resulting matrices are small, the eigenvalues are easily calculated and the bifurcation points are correctly captured. Nine branches of stable and unstable solutions are obtained with this method in an interval of values of the Rayleigh number. Different basis sets are considered in each branch. The reduced basis method permits one to obtain the bifurcation diagrams with much lower computational cost.

中文翻译：

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降基法应用于对流稳定性问题

在许多查询的情况下，数值简化基方法有助于解决参数相关的偏微分方程问题。分叉和不稳定性问题具有这些特征，因为通过改变分叉参数会出现不同的解决方案。通过改变瑞利数，瑞利-贝纳德对流是一个不稳定问题，具有多个稳态解和分支。本文用这种方法解决了相应线性稳定性分析的特征值问题。生成的矩阵很小，特征值易于计算，并且分叉点可以正确捕获。用这种方法在瑞利数的值的间隔中获得了九个稳定和不稳定解的分支。每个分支考虑不同的基础集。