Comparison of ROW, ESDIRK, and BDF2 for Unsteady Flows with the High-Order Flux Reconstruction Formulation

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 the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) solution of the unsteady Navier-Stokes equations. Pseudo-transient continuation is employed to solve the nonlinear equation at each stage of ESDIRK (excluding the first stage) and each step of 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 the linear system at each stage of ROW and each pseudo time iteration of ESDIRK and BDF2. Several numerical experiments, including both laminar and turbulent flow simulations, are conducted to carry out the comparison. We observe that the multistage ROW2 and ESDIRK2 are more efficient than the multistep BDF2, and higher-order implicit time integrators are more efficient than lower-order ones. In general, the ESDIRK method allows a larger physical time step size for unsteady flow simulation than the ROW method when the element-Jacobi preconditioner is employed, especially for wall-bounded flows; and the ROW method can be more efficient than the ESDIRK method when the time step size is refined.

Gust mitigation with an oscillating airfoil at low Reynolds number

“…To solve Eq. ( 9), it can be rewritten as @Q @s ¼ RðQ; rQÞ; (10) and the linearly implicit Rosenbrock-Wanner (ROW) Runge-Kutta method 53 was used for time integration. The general formulation of ROW method is written as follows:…”

Computational Study on the Effects of Unsteady Freestream on an Airfoil Performance at Low Reynolds Numbers

“…The Flux Reconstruction/Correction Procedure via Reconstruction (FR/CPR) [20][21][22][23][24][25][26] method is used to solve the governing equations. A brief introduction of the FR/CPR method is discussed in this section for the sake of completeness, but a full discussion of the method can be found in [20].…”

“…The explicit first stage, singly diagonally implicit Runge-Kutta (ESDIRK) method [25,26] was used for time integration, which is written as…”

A flux reconstruction kinetic scheme for the Boltzmann equation

“…As a further simplification, the explicit singly diagonally implicit Runge-Kutta (ES-DIRK) method employs an explicit first step and thus reduces the degree of the nonlinear systems from SDIRK by one. The comparisons from compressible Navier-Stokes equations indicated that the ESDIRK methods are more efficient than the BDF methods [52,53].…”