Nektar++: Design and implementation of an implicit, spectral/hp element, compressible flow solver using a Jacobian-free Newton Krylov approach

Abstract

At high Reynolds numbers the use of explicit in time compressible flow simulations with spectral/hp element discretization can become significantly limited by time step. To alleviate this limitation we extend the capability of the spectral/hp element open-source software framework, Nektar++, to include an implicit discontinuous Galerkin compressible flow solver. The integration in time is carried out by a singly diagonally implicit Runge-Kutta method. The non-linear system arising from the implicit time integration is iteratively solved by the Jacobian-free Newton Krylov (JFNK) method. A favorable feature of the JFNK approach is its extensive use of the explicit operators available from the previous explicit in time implementation. The functionalities of different building blocks of the implicit solver are analyzed from the point of view of software design and placed in appropriate hierarchical levels in the C++ libraries. In the detailed implementation, the contributions of different parts of the solver to computational cost, memory consumption and programming complexity are also analyzed. A combination of analytical and numerical methods is adopted to simplify the programming complexity in forming the preconditioning matrix. The solver is verified and tested using cases such as manufactured compressible tex, turbulent flow over a circular cylinder at Re = 3900 and shock wave boundary-layer interaction. The results show that the implicit solver can speed-up the simulations while maintaining good simulation accuracy.

Triglobal infinite-wing shock-buffet study

“…The solution converges well, almost monotonically, for Λ = 20 • and 30 • and, interestingly and in contrast to the straight wing, spanwise-uniform flow is found at terminal convergence. At the lower sweep angles of Λ = 5 • and 10 • , convergence stalls, failing to reach the specified tolerance, and different methods, such as selective frequency damping (Åkervik et al 2006) or a stronger implicit Newton-Krylov solver (Yan et al 2021), should be explored in the future to find fully converged base flow in those cases. Close inspection of the corresponding non-converged flow fields suggests indecisiveness in either forming spanwise cellular structures as for sweep angle Λ = 0 • or convergence towards spanwise-uniform flow as for the two largest sweep angles investigated.…”

Development of a Balanced Adaptive Time-Stepping Strategy Based on an Implicit JFNK-DG Compressible Flow Solver

“…In what follows we briefly outline the formulation of a compressible flow solver based on a discontinuous Galerkin (DG) spatial discretization [13], an implicit time integration via ESDIRK temporal discretization schemes [17] and a JFNK method [16]. Further details of the solver can be found in reference [29].…”

“…where w is the quadrature weight, J is the grid metric Jacobian, represents a diagonal matrix, = T (wJ) is the mass matrix, j is the derivative matrix in the jth direction, the superscript represents the corresponding matrices or variables on element boundaries, ̂ n j is a symmetric flux from the SIPG method, is the mapping matrix between and , and is the interpolation matrix from quadrature points of an element to quadrature points of its element boundaries. Details of the spatial discretization can be found in reference [29].…”

“…Implicit time integration schemes have the potential to increase the efficiency of high-Reynolds number simulations by relaxing the challenging time step restriction, and have been successfully adopted in a variety of steady-state and unsteady simulations [2,21,22,27,29]. However, implicit time integration schemes are generally more complex and introduce many parameters, which usually have large influences on the performance of the solver.…”

“…However, implicit time integration schemes are generally more complex and introduce many parameters, which usually have large influences on the performance of the solver. Take the widely used implicit time integration schemes based on Runge-Kutta (RK) temporal discretization scheme and Jacobian-free Newton Krylov (JFNK) method as an example, there are parameters such as the order of RK scheme, the time step size, the Newton iteration convergence tolerance, the linear iteration convergence tolerance and possibly other parameters introduced by preconditioners [15,16,29]. These parameters are usually highly problem dependent and have large influences on the accuracy, efficiency and robustness in specific simulations, the choices of which actually become a complex multi-objective optimization problem.…”