Wednesday 11 Jul 2018: Rayleigh basis method applied to a convective stability Rayleigh-Benard problem and its bifurcations
Francisco Pla - University of Castilla-La Mancha, Spain
Instability and bifurcation problems that require the solution of systems of partial differential equations for a large range of the parameters are computationally expensive and a reduced order method is necessary. The reduced basis approximation is a discretisation method that can be implemented for solving parameter-dependent problems P(f(m); m)=0, where m is the parameter. The method consists of approximating the solution f(m) by a linear combination of appropriate preliminary computed solutions f(m_k) for k=1,2,..N, where the m_k are parameters chosen by an iterative procedure using the Kolmogorov n-width measures [1,2].
Rayleigh-Benard convection problems display multiple steady solutions and bifurcations at different Rayleigh numbers, R. A linear stability analysis of these solutions is performed in  using a spectral collocation method. In  the eigenvalue problem is solved with the reduced basis method. A fixed aspect ratio (G=3.495) is considered and the Rayleigh number varies from 1000 to 2000, where different stable and unstable bifurcation branches are known to appear [3,4]. The reduced basis considered belongs to the eigenfunction spaces from the eigenvalue problems for different types of solutions in the bifurcation diagram. The eigenvalues and eigenfunctions are easily calculated and the bifurcation points are exactly captured. The resulting matrices are small and this allows a drastic reduction of the computational cost.
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