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 Department of Applied Physics
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 Barcelona School of Civil Engineering (ETSECCPB)
 fernando.garciagonzalezestudiant.upc.edu
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A comparison of highorder time integrators for highly supercritical thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Lecture notes in computational science and engineering
Date of publication: 20140110
Journal article
Read the abstract View Share Reference managersThe efficiency of implicit and semiimplicit time integration codes based on backward differentiation and extrapolation formulas for the solution of the threedimensional Boussinesq thermal convection equations in rotating spherical shells was studied in Garcia et al. (J Comput Phys 229:7997¿8010, 2010) at weakly supercritical Rayleigh numbers R, moderate (103) and low (104) Ekman numbers, E, and Prandtl number s = 1. The results presented here extend the previous study and focus on the effect of s and R by analyzing the efficiency of the methods for obtaining solutions at TeX, s = 0. 1 and low and high supercritical R. In the first case (quasiperiodic solutions) the decrease of one order of magnitude does not change the results significantly. In the second case (spatiotemporal chaotic solutions) the differences in the behavior of the semiimplicit codes due to the different treatment of the Coriolis term disappear because the integration is dominated by the nonlinear terms. As in Garcia et al. (J Comput Phys 229:7997¿8010, 2010), high order methods, either with or without time step and order control, increase the efficiency of the time integrators and allow to obtain more accurate solutions.
The efficiency of implicit and semiimplicit time integration codes based on backward differentiation and extrapolation formulas for the solution of the threedimensional Boussinesq thermal convection equations in rotating spherical shells was studied in [5] at weakly supercritical Rayleigh numbers R, moderate (10−3) and low (10−4) Ekman numbers, E, and Prandtl number = 1. The results presented here extend the previous study and focus on the effect of and R by analyzing the efficiency of the methods for obtaining solutions at E = 10−4, = 0.1 and low and high supercritical R. In the first case (quasiperiodic solutions) the decrease of one order of magnitude does not change the results significantly. In the second case (spatiotemporal chaotic solutions) the differences in the behavior of the semiimplicit codes due to the different treatment of the Coriolis term disappear because the integration is dominated by the nonlinear terms. As in [5], high order methods, either with or without time step and order control, increase the efficiency of the time integrators and allow to obtain more accurate solutions.
Postprint (author’s final draft) 
Exponential versus IMEX highorder time integrators for thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Bonaventura, Luca; Net Marce, Marta; Sanchez Umbria, Juan Jose
Journal of computational physics
Date of publication: 20140501
Journal article
Read the abstract View Share Reference managersWe assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the threedimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicitexplicit (IMEX) multistep methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earth's outer core conditions. (C) 2014 Elsevier Inc. All rights reserved.
We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the threedimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicit–explicit (IMEX) multistep methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earthʼs outer core conditions. 
Numerical simulations of thermal convection in rotating spherical shells under laboratory conditions
Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose; Net Marce, Marta
Physics of the Earth and planetary interiors
Date of publication: 20140305
Journal article
Read the abstract View Share Reference managersAn exhaustive study, based on numerical threedimensional simulations, of the Boussinesq thermal convection of a fluid confined in a rotating spherical shell is presented. A moderately low Prandtl number fluid (ro = 0.1) bounded by differentiallyheated solid spherical shells is mainly considered. Asymptotic power laws for the mean physical properties of the flows are obtained in the limit of low Rossby number and compared with laboratory experiments and with previous numerical results computed by taking either stressfree boundary conditions or quasigeostrophic restrictions, and with geodynamo models. Finally, using parameters as close as possible to those of the Earth's outer core, some estimations of the characteristic time and length scales of convection are given. © 2014 Elsevier B.V. 
Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a doubleHopf bifurcation
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
Physical review E: statistical, nonlinear, and soft matter physics
Date of publication: 20130322
Journal article
Read the abstract Access to the full text Share Reference managersA methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, NewtonKrylov, and Arnoldi methods. An application to the study of a doubleHopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=104 and =0.1, respectively, radius ratios n[0.32,0.35], Rayleigh numbers R[1.8105,6105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.
A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, NewtonKrylov, and Arnoldi methods. An application to the study of a doubleHopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=10−4 and σ=0.1, respectively, radius ratios η∈[0.32,0.35], Rayleigh numbers R∈[1.8×105,6×105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves. 
Simulations of thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Bonaventura, Luca; Net Marce, Marta; Sanchez Umbria, Juan Jose
SIAM Conference on Mathematical Computational Issues in the Geosciences
Presentation's date: 20130617
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Time integration of thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Bonaventura, Luca; Net Marce, Marta; Sanchez Umbria, Juan Jose
SemiLagrangian Day Workshop
Presentation's date: 20130206
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Bifurcations from travelling waves in radially heated rotating spherical shells
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
Reunión Científica sobre Nuevas Técnicas Numéricas para Problemas No Estacionarios de Fluidos
Presentation's date: 20130217
Presentation of work at congresses
Read the abstract View Share Reference managersTravelling waves appearing in the thermal convection of a pure fluid contained in a spherical shell with the boundaries at different temperatures are studied. They are computed, by using continuation methods, as steady solutions of a system for the waves, in the frame of reference of the spheres. NavierStokes equations are written in terms of two scalar potentials for the velocity, which are expanded, as the temperature, in spherical harmonics, and collocation is employed in the radius. The special blocktridiagonal structure of the linear part of the equations provides a preconditioner, which allows an efficient calculation of the waves. Their stability is also studied, and the secondary bifurcations to subharmonic or modulated waves are detected. 
Numerical study of the onset of thermosolutal convection in rotating spherical shells
Net Marce, Marta; Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose
Physics of fluids
Date of publication: 20120601
Journal article
Read the abstract Access to the full text Share Reference managersThe influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored. 
A comparison of highorder time integrators for highly supercritical thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
International Conference on Spectral and HighOrder Methods
Presentation's date: 20120628
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Thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando
Defense's date: 20121130
Department of Applied Physics, Universitat Politècnica de Catalunya
Theses
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Dinámica supercrítica en la convección térmica de un fluido en geometría esférica en rotación
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Congreso de Ecuaciones Diferenciales y Aplicaciones y Congreso de Matemàtica Aplicada
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Stability analysis for the onset of convection in rotating fluid binary mixtures in spherical shells
Net Marce, Marta; Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose
European Nonlinear Oscillations Conferences
Presentation's date: 201107
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Azimuthal waves and their stability in externally heated rotating spherical shells
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
European Nonlinear Oscillations Conferences
Presentation's date: 201107
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Secondary bifurcations in radially heated rotating spherical shells
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
International Congress on Industrial and Applied Mathematics
Presentation's date: 20110719
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Bifurcations from travelling waves in radially heated rotating spherical shells
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
Coherent Structures in Dynamical Systems
Presentation's date: 201105
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Azimuthal waves, their stability and connecting orbits in thermal convection in rotating spherical shells
Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando; Net Marce, Marta
Computational Methods in Dynamics
Presentation's date: 20110708
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Cálculo numérico de variedades invariantes en EDPS disipativas. Aplicaciones a la convección térmica
Net Marce, Marta; Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose
Participation in a competitive project
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A comparison of highorder time integrators for thermal convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; GarciaArchilla, Bosco; Sanchez Umbria, Juan Jose
Journal of computational physics
Date of publication: 20101001
Journal article
Read the abstract Access to the full text Share Reference managersA numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semiimplicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.
Postprint (author’s final draft) 
Estudio comparativo de la eficiencia de diversos integradores para la simulación numérica de la convección térmica en geometría esférica y rotación
Garcia Gonzalez, Fernando; Net Marce, Marta; GarciaArchilla, Bosco; Sanchez Umbria, Juan Jose
Congreso de Ecuaciones Diferenciales y Aplicaciones / Congreso de Matemática Aplicada
Presentation's date: 20090924
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersThe efficiency of implicit and semiimplicit time integration codes based on bac kward differentiation and extrapolation formulae for the solution of the threedimensio nal Boussinesq thermal convection equations in rotating spherical shells is studied. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The results show that high order methods, either with or without time step and order control, increase the efficiency of the time integrators, and allow to obtain more accurate solutions. 
Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high Taylor numbers
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Physical review letters
Date of publication: 20081103
Journal article
Read the abstract Access to the full text Share Reference managersThe onset of thermal convection in a rotating spherical shell of intermediate radius ratio ¼ 0:4 is studied numerically for Taylor numbers Ta 1011 and the Prandtl number of the liquid sodium ( ¼ 0:01). For the first time, it is shown that at very high Taylor numbers the first unstable mode can be antisymmetric with respect to the equator and confined inside a cylinder tangent to the inner sphere at the equator (polar mode). The exponent of the power law determined from the asymptotic dependence of the critical Rayleigh number for very high Ta is 0.57, lower than 2=3, given theoretically for the spiraling columnar modes, and than 0.63, found numerically for the outer equatorially attached modes 
Estudio comparativo de la eficiencia de diversos integradores para la simulación numérica de la convección térmica en geometría esférica en rotación
Garcia Gonzalez, Fernando; Net Marce, Marta; GarciaArchilla, Bosco; Sanchez Umbria, Juan Jose
Congreso No Lineal
Presentation's date: 20080616
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Numerical techniques for the onset of thermal convection in spherical geometry
Net Marce, Marta; Sanchez Umbria, Juan Jose; Garcia Gonzalez, Fernando
SIAM Conference on Applications of Dynamical Systems
Presentation's date: 2007
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A note on Weak Stability Boundaries
Garcia Gonzalez, Fernando; Gómez Muntané, Gerard
Celestial mechanics and dynamical astronomy
Date of publication: 20061208
Journal article
Read the abstract View Share Reference managersThis paper is devoted to clarify the algorithmic definition of the weak stability boundary in the framework of the planar Restricted Three Body Problem. The role of the invariant hyperbolic manifolds associated to the central manifolds of the libration points L1 and L2, as boundary of the weak stability region, is shown 
Low Prandtl number convection in rotating spherical shells
Garcia Gonzalez, Fernando; Sanchez Umbria, Juan Jose; Net Marce, Marta
Carles Simó Fest (60th birthday). Conference on Dynamical Systems
Presentation's date: 20060529
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Low Prandtl number convection in rotating spherical shells
Garcia Gonzalez, Fernando; Net Marce, Marta; Sanchez Umbria, Juan Jose
Novena Trobada Matemàtica de la Societat Catalana de Matemàtiques
Presentation's date: 20060422
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