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Aligned numerical methods for anisotropic elliptic problems in bounded domains for plasma edge simulations

Abstract : Highly anisotropic elliptic problems occur in many physical models that need to be solved numerically. In the problems investigated in this thesis, a direction of dominant diffusion exists (called here parallel direction), along which the diffusivity is several orders of magnitude larger than in the perpendicular direction. In this case, standard finite-difference methods are generally not designed to provide an optimal discretization and may lead to the perpendicular diffusion being artificially supplemented by a potentially large contribution stemming from errors in approximating parallel diffusion. This thesis focuses on three main axes to suitably solve anisotropic elliptic equations: an aligned, conservative finite-difference scheme to discretize the Laplacian operator, a reformulated Helmholtz equation to avoid spurious numerical diffusion, and a solver based on multigrid methods as a preconditioner of GMRES routine. Although the scope of this thesis is the application on plasma edge physics, results are relevant to any highly anisotropic model flow in bounded domains. In Chapter 1, a short introduction to magnetically confined fusion is presented identifying the numerical problems raised by solving fluid equations, in particular in the Scrape-Off Layer region. The numerical problem which is dealt with is an anisotropic elliptic problem where diffusivity is 5 to 8 orders of magnitude larger in the parallel direction. This large parallel diffusivity results in long wavelengths in the parallel direction, a central characteristic to the understanding of methods discussed in this thesis. In Chapter 2, a bibliographic introduction to numerical methods dedicated to the solution of anisotropic elliptic equations is presented, with a focus on finite-difference methods. Aligned methods, and their potential to compute solutions with accuracy comparable to standard methods with much lower number of mesh points, are presented. In Chapter 3 we propose an original aligned discretization scheme using non-aligned Cartesian grids. Based on the Support Operator Method, the self-adjointness of the parallel diffusion operator is maintained at the discrete level. Compared with existing methods, the present formulation further guarantees the conservativity of the fluxes in both parallel and perpendicular directions. For bounded domains, a discretization of boundary conditions is presented ensuring comparable accuracy of the solution. Numerical tests based on manufactured solutions show that the method provides accurate and stable numerical approximations in both periodic and bounded domains with a drastically reduced number of degrees of freedom with respect to non-aligned approaches. A reformulation of the Helmholtz equation is presented in Chapter 4 to limit spurious numerical diffusion. The method is based on splitting of the original problem into two distinct problems for the aligned and the non-aligned parts of the solution. These two contributions are separated by filtering methods which are evaluated. Tests cases showthis reformulation eliminates spurious perpendicular diffusion, with larger impact on accuracy with higher parallel diffusivities.Finally, with the aim of solving elliptic anisotropic equations for large systems efficiently, a geometric multigrid algorithm is proposed in Chapter 5 in bounded domains. The algorithm scales adequately with the number of degrees of freedom, and shows a clear advantage upon standard iterative methods when the parallel diffusivity is very large. This algorithm is later posed as preconditioner of a GMRES solver, finding computationally efficient algorithm compared with direct solvers solving elliptic equations under any boundary conditions.The thesis is concluded by a critical analysis of the numerical aspects of aligned discretizations investigated. Special attention is given to the application of the investigated schemes in 3D plasma turbulence codes, such as the TOKAM3X developed by CEA.
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Submitted on : Friday, May 15, 2020 - 3:39:09 PM
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Juan Antonio Soler Vasco. Aligned numerical methods for anisotropic elliptic problems in bounded domains for plasma edge simulations. Numerical Analysis [math.NA]. Ecole Centrale Marseille, 2019. English. ⟨NNT : 2019ECDM0005⟩. ⟨tel-02591943⟩



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