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

## TL;DR

This paper compares Jacobian-free Rosenbrock-Wanner, ESDIRK, and BDF2 methods for unsteady flow simulations with high-order flux reconstruction, highlighting efficiency and stability differences.

## Contribution

It provides a comprehensive comparison of these methods, including implementation details and performance analysis for high-order unsteady flow simulations.

## Key findings

- ROW and ESDIRK are more efficient than BDF2.
- ROW methods can outperform ESDIRK in efficiency.
- ESDIRK allows larger time steps than ROW.

## 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.