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Wednesday 11 Jul 2018Rayleigh basis method applied to a convective stability Rayleigh-Benard problem and its bifurcations

Francisco Pla - University of Castilla-La Mancha, Spain

H103 16:00-17:00

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 [3] using a spectral collocation method. In [4] 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.

[1] Y. Maday, A.T. Patera and G. Turinici, Convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput., Vol. 7(1-4), pp. 437-446, 2002.

[2] C. Prud’homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced basis output bound methods, Journal of Fluids Engineering, Vol 124(1), pp. 70-80, 2002.

[3] F. Pla, A.M. Mancho and H. Herrero, Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio. Physica D, Vol 238, pp. 572-580, 2009.

[4] H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Benard). Computer Methods in Applied Mechanics and Engineering, Vol. 261-262, pp. 132-141, 2013.

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