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Analysis of time-domain integration methods for the simulation of thermal convection in an annulus

Abstract : Numerical simulations of outer core thermal convection of the Earth have been an essential tool in understanding the dynamics of magnetic field generation which surrounds the Earth. Efficient numerical strategies to solve this system of governing equations are of interest in the community of deep Earth research because, current numerical geodynamo models are on the quest to operate at the actual parameters of the Earth. There are many avenues for the improvement of the numerical model. In this thesis, we focus on the time domain integration techniques for solving such problems so that we may push the parameter boundaries further. We solve for a thermal convection problem in a 2D annulus. We use a pseudospectral method for spatial discretization. With respect to the time discretization part, the governing equations contain both numerically stiff (diffusive) and non-stiff (advective) components. A common practice is to treat the diffusive part implicitly and the advective part explicitly so as to alleviate the timestep restriction which happens when we use a purely explicit method. These are known as the IMEX time integrators. We focus on these IMEX methods and analyze their performance when applied to this problem. We consider two families of IMEX methods, the multistep methods and the multistage IMEX Runge-Kutta methods (IMEX-RK). We do a systematic survey of input parameters namely the Rayleigh number (Ra) and the Prandtl number (Pr), which control the thermal forcing and the ratio of momentum to thermal diffusivities respectively. Our focus is on the strongly nonlinear flow regimes and we observe that, as the Reynolds number increases, few of the IMEX-RK methods perform better than multistep methods. Specifically, we compare the performances of the IMEX-RK methods with the second order Crank-Nicholson and Adams-Bashforth (CNAB2) method, which is widely used in the geodynamo community. We find some of the higher order methods to perform better than CNAB2 for large Reynolds numbers. This result opens up the possibility of utilizing such higher order methods for the full 3D dynamo calculations. However, in most other cases, multistep methods of a given order outperform IMEX-RK methods of the same order.
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Venkatesh Gopinath. Analysis of time-domain integration methods for the simulation of thermal convection in an annulus. Earth Sciences. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCC035⟩. ⟨tel-02612607⟩

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