Implicit time-stepping for scale-resolved industrial flow simulations using spectral/HP elements

Accurate numerical simulations of unsteady fluid flows require scale-resolving simulations which are, however, limited in applications due to high computational costs. Since computational resources are limited, cost and time constraints require algorithmic improvements instead. Scale-resolving simulations typically rely on explicit time-stepping schemes due to low computational cost per time step. In this work, we investigate an implicit time-stepping scheme that increases the computational cost for each time step, but permits larger time steps to reduce the time-to-solution. In particular, we study a velocity correction scheme that is commonly employed for solving the incompressible Navier-Stokes equations. The semi-implicit velocity correction scheme uses implicit diffusion and explicit advection treatment. The explicit advection treatment becomes a performance bottleneck for high Reynolds number flows around complex geometries, because of a CFL-type condition that constrains the algorithm's stability to small time steps. We explore a linear-implicit velocity correction scheme which removes the CFL limitation. The linear-implicit scheme uses a linearisation of the advection operator and, hence, preserves the linear structure of the semi-implicit algorithm. We perform a comparison of both velocity correction schemes and look into their stability, accuracy and computational performance. Our investigation includes canonical problems and a two-dimensional cylinder flow for verification. Further, we investigate turbulent flows at high Reynolds number and around challenging geometries based on Formula 1 race cars. We find that the linear-implicit scheme allows strong improvements in the stability allowing up to 100-times larger time step sizes on the most complex 3D geometry. The influence on the accuracy with larger time step sizes varies and all cases show negligible differences up to at least 10-times larger time step sizes. Additionally, the stability improvement provides leverage for the computational performance enabling 2-fold speed-up with minor loss in accuracy and 10-fold speed-up in time-to-solution with stronger losses in the accuracy.