Application of a line implicit method to fully coupled system of equations for turbulent flow problems

For unstructured finite volume methods, we present a line implicit Runge–Kutta method applied as smoother in an agglomerated multigrid algorithm to significantly improve the reliability and convergence rate to approximate steady-state solutions of the Reynolds-averaged Navier–Stokes equations. To describe turbulence, we consider a one-equation Spalart–Allmaras turbulence model. The line implicit Runge–Kutta method extends a basic explicit Runge–Kutta method by a preconditioner given by an approximate derivative of the residual function. The approximate derivative is only constructed along predetermined lines which resolve anisotropies in the given grid. Therefore, the method is a canonical generalisation of point implicit methods. Numerical examples demonstrate the improvements of the line implicit Runge–Kutta when compared with explicit Runge–Kutta methods accelerated with local time stepping