[Paper Review] A comparative study of implicit Jacobian-free Rosenbrock-Wanner, ESDIRK and BDF methods for unsteady flow simulation with high-order flux reconstruction formulations

Lai Wang, Meilin Yu|arXiv (Cornell University)|Apr 9, 2019

Computational Fluid Dynamics and Aerodynamics42 references8 citations

TL;DR

This study compares Jacobian-free Rosenbrock-Wanner (ROW), ESDIRK, and BDF2 methods for unsteady flow simulations using high-order flux reconstruction (FR/CPR) schemes. Using a GMRES solver with element-Jacobi preconditioning and pseudo-transient continuation, the authors find that ROW and ESDIRK methods are more efficient than BDF2, with ROW potentially outperforming ESDIRK when linear solver tolerances are tight, though ESDIRK allows larger time steps and is more robust under looser tolerances.

ABSTRACT

We conduct a comparative study of the Jacobian-free linearly implicit Rosenbrock-Wanner (ROW) methods, the explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) methods, and the second-order backward differentiation formula (BDF2) for unsteady flow simulation using spatially high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) formulations. The pseudo-transient continuation is employed to solve the nonlinear systems resulting from the temporal discretizations with ESDIRK and BDF2. A Jacobian-free implementation of the restarted generalized minimal residual method (GMRES) solver is employed with a low storage element-Jacobi preconditioner to solve linear systems, including those in linearly implicit ROW methods and those from linearization of the nonlinear systems in ESDIRK and BDF2 methods. We observe that all ROW and ESDIRK schemes (from second order to fourth order) are more computationally efficient than BDF2, and ROW methods can potentially be more efficient than ESDIRK methods. However, the convergence tolerance of the GMRES solver for ROW methods needs to be sufficiently tight to preserve the nominal order of accuracy. In general, ESDIRK methods allow a larger physical time step size for unsteady flow simulation than ROW methods do.

Motivation & Objective

• To evaluate the accuracy and computational efficiency of Jacobian-free ROW, ESDIRK, and BDF2 time integration schemes for high-order FR/CPR discretizations.
• To investigate the impact of linear and nonlinear solver tolerances on the performance of ROW and ESDIRK methods in unsteady flow simulations.
• To assess the robustness and order of accuracy preservation of these methods under varying flow complexity, particularly in wall-bounded turbulent flows.
• To examine the role of preconditioning and Krylov subspace solvers in the efficiency of implicit time integration for high-order CFD methods.
• To determine the optimal balance between temporal accuracy, stability, and computational cost in long-time unsteady simulations.

Proposed method

• Employed high-order flux reconstruction (FR/CPR) for spatial discretization of the Navier-Stokes equations.
• Applied multistage time integration schemes: ROW (2nd–4th order), ESDIRK (2nd–4th order), and BDF2 for temporal discretization.
• Used a Jacobian-free GMRES solver with a low-storage element-Jacobi preconditioner to solve linear systems in all methods.
• Implemented pseudo-transient continuation to solve nonlinear systems arising in ESDIRK and BDF2 methods.
• Tightened the linear solver tolerance for ROW to preserve nominal order of accuracy, while using looser tolerances for ESDIRK and BDF2 to reduce cost.
• Conducted convergence studies using the Taylor-Green vortex decay problem with reference spectral solution for error assessment.

Experimental results

Research questions

• RQ1 How do the computational efficiencies of ROW, ESDIRK, and BDF2 compare when solving unsteady flows with high-order FR/CPR schemes?
• RQ2 What is the impact of linear and nonlinear solver tolerances on the accuracy and efficiency of ROW and ESDIRK methods?
• RQ3 Can ROW methods maintain their nominal order of accuracy when using inexact linear solves, and how does this affect overall efficiency?
• RQ4 How do the time step sizes allowed by ROW and ESDIRK compare in unsteady flow simulations, particularly for complex flows like wall-bounded turbulence?
• RQ5 To what extent does the performance of the element-Jacobi preconditioner affect the efficiency of ROW methods in high-order simulations?

Key findings

• ROW and ESDIRK methods are significantly more computationally efficient than BDF2 for unsteady flow simulations with high-order FR/CPR schemes.
• When the linear solver tolerance for ROW is tightly controlled (e.g., 10−6), ROW methods are more efficient than ESDIRK methods due to better convergence per time step.
• ESDIRK methods allow larger physical time steps than ROW methods, improving overall efficiency when solver tolerances are relaxed.
• The convergence tolerance of the GMRES solver for ROW methods must be sufficiently tight (e.g., 10−6) to preserve the nominal order of accuracy; looser tolerances lead to order reduction.
• ESDIRK methods are more robust than ROW methods under varying flow conditions, such as unsteady flows over walls, due to more consistent performance and reduced sensitivity to tolerance settings.
• The performance of the element-Jacobi preconditioner significantly affects ROW efficiency, limiting the time step size for good convergence behavior, whereas ESDIRK remains effective at larger time steps.