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

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.

A dynamically load-balanced parallel $ p $-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation

“…The second-order, three-stage ESDIRK2 [51], and fourth-order, six-stage ESDIRK4 [61] methods are employed in this paper. A comparative study of different implicit time integration methods can be found in [53]. In every stage except the first one, a nonlinear system is to be solved, which can be expressed as…”

“…The pseudo transient continuation is an inexact Newton's method. Therefore, we assign a relatively large tolerance to the GMRES solver, i.e., tol gmres rel = 10 −1 , to save computational cost [53]. However, for stiff problems, tol gmres rel = 10 −1 may lead to divergence of the pseudo transient continuation [64].…”

“…Diagonally implicit RK methods [51] and backward differentiation formula (BDF) methods are among the most popular implicit time integration methods. Recently, linearly implicit Rosenbrock methods have become popular for under-resolved turbulence simulation [52,53]. Matrix-based implicit methods are notorious for the large memory consumption.…”

“…Matrix-based implicit methods are notorious for the large memory consumption. Therefore, the matrix-free implementation [53,54] is usually employed to reduce memory usage [55] for massive turbulence simulation. Though Rosenbrock methods can be potentially more efficient than ESDIRK methods [53], we employ ESDIRK methods in this study since they are more robust than Rosenbrock methods in the context of matrix-free implementation with an element-Jacobi preconditioner.…”

A High Order Flux Reconstruction Interface Tracking Method Using Preconditioned Phase Field

“…Recent trends in CFD research indicate a steadily increasing interest in high-order numerical methods. The ability of such methods to produce results with more accuracy on coarser grids when compared to conventional low order methods resulted in a growing consensus among CFD practitioners that high-order methods may constitute the basis of next-generation CFD research tools [22,23]. In line with this vision, some efforts in multi-phase CFD research aimed at using the superior numerical properties of high-order methods to address some of the limitations of level set and VOF approaches.…”