Semi-Lagrangian discontinuous Galerkin methods for fluid and kinetic applications
Date
2020
Authors
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Publisher
University of Delaware
Abstract
The challenges in performing complex large-scale realistic experiments such as tokamak fusion test lead to the extensive study and development of high-fidelity computational tools. Among them, high order accurate schemes are essential components in scientific computing area owing to their high efficiency and high resolution properties. The objective of this PhD dissertation is to investigate efficient, highly accurate and stable numerical methods, especially to apply the semi-Lagrangian (SL) discontinuous Galerkin (DG) method to solving convection diffusion problems and stiff relaxation problems. ☐ The first contribution of this work is an efficient high order SL DG method for solving linear convection diffusion equations. The method generalizes the SL DG method for transport equations, making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta (RK) methods together with source terms. The proposed method is named the `SL DG-LDG' method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order spatial and temporal accuracy, compactness, mass conservation, and stability under large time stepping size. An $L^2$ stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method is demonstrated by a group of numerical tests in one and two dimensions. ☐ The second one is that we study the asymptotic accuracy property of time integrators in the SL setting when applied to a class of stiff relaxation problems, especially towards the hydrodynamic regimes. A SL nodal DG (NDG) method is also proposed to the kinetic model. In particular, we perform accuracy analysis in order to achieve the asymptotic accuracy property of SL-diagonally implicit RK (DIRK) methods. An extra order condition for asymptotic third order accuracy is derived in the limiting fluid regime, based on which a family of third order asymptotically accurate DIRK time discretization methods are developed. Stability of the newly proposed third order DIRK methods are studied via Von Neumann analysis to a simplified linear two-velocity kinetic model. As an application of our theoretical analysis, we couple the SL NDG method with DIRK methods and apply asymptotically accurate SL NDG-DIRK schemes to the BGK model. The spatial discretization is performed by a mass conservative SL NDG method, while the temporal discretization of the stiff relaxation term is realized by stiffly accurate DIRK methods along characteristics. A local maximum principle preserving limiter is added to control numerical oscillations in the transport step. Thanks to the SL and implicit nature of time discretizations, the time stepping constraint is relaxed and much larger than that from an Eulerian framework with explicit treatment of the collision term. Extensive numerical tests are presented to verify the high order spatial and temporal accuracy, mass conservation and asymptotic accuracy of the proposed schemes.
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Keywords
Semi-Lagrangian discontinuous Galerkin methods, Convection diffusion, Transport equations