Abstract
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.
Highlights
Bifurcations and instabilities in differential equations are features that allow the explanation of many fluid dynamics phenomena in nature and industrial processes [1]
The bifurcation points are correctly captured with the reduced basis method looking at the intersection of branches of solutions or regarding the eigenvalues of the linear stability analysis
The relative errors on the solutions in the unstable branches are of the same order as the stable ones, O(10–2), and O(10–4) after a post-processing, and become worse near the bifurcation points
Summary
Bifurcations and instabilities in differential equations are features that allow the explanation of many fluid dynamics phenomena in nature and industrial processes [1]. An example is the Rayleigh–Bénard convection problem [2, 3]. Rayleigh–Bénard and related natural convection phenomena are usual in many industrial applications. In the formation of microstructures during the cooling of molten metals in computer chips or large scale equipments. The model equations in this case are the incompressible Navier– Stokes equations coupled with a heat equation under the Boussinesq approximation. The conductive solution becomes unstable for a critical vertical temperature gradient beyond a certain threshold and a convective motion sets in, and, depending on boundary conditions and other external physical parameters, new convective patterns occur [1]
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