Linearly implicit rosenbrock methods for unsteady flow simulations

In this work, we present details on the implementation of linear Rosenbrock-type methods and their application to unsteady flow problems. In recent years, especially in the field of high-order methods, diagonally implicit Runge-Kutta (DIRK) methods have gained popularity. However, linear Rosenbrock methods are an easier-to-implement and −to-use alternative. Their implementation requires little more than what is already commonly available in most flow solvers. We present results covering direct numerical simulation (DNS), large-eddy simulation (LES), and unsteady Reynolds-averaged Navier-Stokes (URANS) applications and show that when combined with a second-order finite volume method, second-order Rosenbrock schemes provide sufficient accuracy. In particular, the two-stage second-order scheme (ROS2) shows simultaneously increased performance and accuracy compared to the popular second-order backward differentiation formula (BDF2). At the same time, it consistently outperforms similar DIRK schemes in our implementation. As an addition to a legacy finite volume (FV) solver, Rosenbrock methods can increase performance and accuracy and extend its range of applicability and lifespan.