The paper presents the development and investigation of an adaptive-smoothing (AS) procedure in conjunction with the full multigrid–full approximation storage method. The latter has been previously implemented by the authors  for the incompressible Navier–Stokes equations in conjunction with the artificial-compressibility method and forms the basis for investigating the current AS approach. The principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and, subsequently, to reduce the total computing time. The implementation of the AS approach is investigated in conjunction with three different adaptivity criteria for two- and three-dimensional incompressible flows. Furthermore, a dynamic procedure (henceforth labeled dynamic adaptivity) for defining variably the AS parameters during the computation is also proposed and its performance is investigated in contrast to AS with constant parameters (henceforth labeled static adaptivity). Both static and dynamic adaptivity can provide similar acceleration, but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method. Numerical experiments are presented for attached and separated flows around airfoils as well as for three-dimensional flow in a curved channel. For external flows the AS performs better when it is applied in all grid levels of the multigrid method, while for internal flows the best performance is achieved when AS is applied in the finest grid only. Overall, the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing.
- Navier–Stokes equations