Implicit velocity correction schemes for scale-resolving simulations of incompressible flow: Stability, accuracy, and performance

H. Wüstenberg, A. Liosi, S. J. Sherwin, J. Peiró, D. Moxey

Comput. Phys. Commun., vol. 331, pp. 110473 (2027)

@article{wustenberg-2027,
  title = {Implicit velocity correction schemes for scale-resolving simulations of incompressible flow: Stability, accuracy, and performance},
  journal = cpc,
  volume = {331},
  pages = {110473},
  year = {2027},
  doi = {10.1016/j.cpc.2026.110473},
  url = {https://www.sciencedirect.com/science/article/pii/S0010465526004546},
  author = {Wüstenberg, H. and Liosi, A. and Sherwin, S. J. and Peiró, J. and Moxey, D.},
  abstract = {Scale-resolving simulations of high-Reynolds-number incompressible flow are often limited by the Courant–Friedrichs–Lewy (CFL) restriction of explicit time-stepping schemes, leading to small steps and long times-to-solution. We systematically compare two implicit formulations of the velocity correction scheme – a linear-implicit approach and a sub-stepping (semi-Lagrangian) method – with a standard semi-implicit formulation in a high-order spectral/hp element framework. The schemes are assessed in terms of stability limits, temporal accuracy, and computational performance for implicit large-eddy simulation of the Imperial Front Wing benchmark, a complex high Reynolds number geometry with curved surfaces that imposes strict CFL constraints. The sub-stepping scheme extends the stable time step size by a factor of 20, and the linear-implicit scheme by up to a factor of 100 when equal-order velocity and pressure spaces are used. While increasing the cost per time step, the linear-implicit scheme reduces the time-to-solution by up to a factor of 8.7 at this largest step, whereas the sub-stepping scheme reduces it by only 12\%. With application targets of 1\% for mean lift and 5\% for mean drag, the fastest accepted linear-implicit case reduces time-to-solution by a factor of 1.53 at ten times the CFL-limited step. The main-plane separation bubble is already substantially shortened at this step, showing that force agreement can mask changes in transition-sensitive flow structure. The results quantify the trade-off between stability, accuracy, and computational cost for implicit velocity correction schemes on complex geometries and provide guidance for selecting time integration strategies in large-scale scale-resolving simulations.}
}

Scale-resolving simulations of high-Reynolds-number flow are usually limited by the CFL condition, which forces a small time step and a long time to solution. This paper compares two implicit forms of the velocity correction scheme, a linear-implicit one and a sub-stepping semi-Lagrangian one, against the standard semi-implicit formulation, on an implicit large-eddy simulation of the Imperial Front Wing. It measures what each buys in stable step size and in wall-clock time, and finds that agreement in the forces can hide changes in the flow structure underneath.