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

Yu Pan,Zhen-Guo Yan,Joaquim Peiró,Spencer J Sherwin

doi:10.1007/s42967-021-00138-1

Abstract

A balanced adaptive time-stepping strategy is implemented in an implicit discontinuous Galerkin solver to guarantee the temporal accuracy of unsteady simulations. A proper relation between the spatial, temporal and iterative errors generated within one time step is constructed. With an estimate of temporal and spatial error using an embedded Runge-Kutta scheme and a higher order spatial discretization, an adaptive time-stepping strategy is proposed based on the idea that the time step should be the maximum without obviously influencing the total error of the discretization. The designed adaptive time-stepping strategy is then tested in various types of problems including isentropic vortex convection, steady-state flow past a flat plate, Taylor-Green vortex and turbulent flow over a circular cylinder at {Re}=3,900. The results indicate that the adaptive time-stepping strategy can maintain that the discretization error is dominated by the spatial error and relatively high efficiency is obtained for unsteady and steady, well-resolved and under-resolved simulations.

Highlights

Implicit time integration schemes have the potential to increase the efficiency of highReynolds 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]
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]
The basic idea behind the time step adaptation strategy proposed in Sect. 2.2 is the relation between temporal and spatial errors that is sufficient for maintaining temporal accuracy in unsteady simulations

Summary

Introduction

Implicit time integration schemes have the potential to increase the efficiency of highReynolds 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]. 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 become a complex multi-objective optimization problem. Lian et al [18] addressed this problem with the solution-limiting method and claimed that the robustness is largely improved Other attempts on this topic are mainly some techniques such as rolling back and recomputing with a smaller time step if the solution is not satisfactory [30]. Significant progress has been made, as is concluded in reference [6] optimal CFL evolution is still an open problem

Results

Discussion

Conclusion