Efficient implicit spectral/hp element DG techniques for compressible flows

Abstract

In the simulation of stiff problems, such as fluid flows at high Reynolds numbers, the efficiency of explicit time integration is significantly limited by the need to use very small time steps. To alleviate this limitation and to accelerate compressible flow simulations based on high-order spectral/$hp$ element methods, an implicit time integration method is developed using singly diagonally implicit Runge-Kutta temporal discretization schemes combined with a Jacobian-free Newton Krylov (JFNK) method. This thesis studies several topics influencing the efficiency, accuracy and robustness of the solver.

Firstly, an efficient partially matrix-free block relaxed Jacobi (BRJ) preconditioner is proposed, in which the Jacobian matrix and preconditioning matrices are properly approximated based on studies of their influences on convergence. The preconditioner only forms and stores the diagonal part of the Jacobian matrix while the off-diagonal operators are calculated on the fly. Used together with techniques such as using single precision data, the BRJ can largely reduce the memory consumption when compared with matrix-based ones like incomplete LU factorization preconditioners (ILU). To further accelerate the solver, influences of different parts of the flux Jacobian on the preconditioning effects are studied and terms with minor influences are neglected. This reduces the computational cost of the BRJ preconditioner by about 3 times while maintaining similar preconditioning effects.

Secondly, adaptive strategies for a suitable choice of some free parameters are designed to maintain temporal accuracy and relatively high efficiency. The several free parameters in the implicit solver have significant influences on the accuracy, efficiency and stability. Therefore, designing proper strategies in choosing them is essential for developing a robust general purpose solver. Based on the idea of constructing proper relations between the temporal, spatial and iterative errors, adaptive strategies are designed for determining the time step and Newton tolerance. These parameters maintain temporal accuracy of the solver in the sense that further decreasing temporal and iterative errors will not obviously improve the efficiency. Meanwhile, they maintain relatively efficient by avoiding excessively small time step and Newton tolerance. The strategies are tested in different types of cases to illustrate their performance and generality.

Finally, the implicit solver is studied in high-fidelity simulations of turbulent flows based on a hierarchical implementation in the open-source spectral/$hp$ element framework Nektar++. The solver is applied to large-eddy simulations of Taylor-Green vortex flow, turbulent channel flow and flow over a circular cylinder cases. The efficiency of the solver and the prediction accuracy of these problems are studied. The results show that the solver yields good predictions in turbulence simulations whilst keeping good stability and high efficiency.