Convergence of defect-correction and multigrid iterations for inviscid flows

Abstract

Convergence of multigrid and defect-correction iterations is comprehensively studied within different incompressible and compressible inviscid regimes on high-density grids. Good smoothing properties of the defectcorrection relaxation have been shown using both a modified Fourier analysis and a more general idealizedcoarse-grid analysis. Single-grid defect correction alone has some slowly converging iterations on grids of medium density. The convergence is especially slow for near-sonic flows and for very low compressible Mach numbers. Additionally, the fast asymptotic convergence seen on medium density grids deteriorates on highdensity grids. Certain downstream-boundary modes are very slowly damped on high-density grids. Multigrid scheme accelerates convergence of the slow defect-correction iterations to the extent determined by the coarsegrid correction. The two-level asymptotic convergence rates are stable and significantly below one in most of the regions but slow convergence is noted for near-sonic and very low-Mach compressible flows. Multigrid solver has been applied to the NACA 0012 airfoil and to different flow regimes, such as near-tangency and stagnation. Certain convergence difficulties have been encountered within stagnation regions. Nonetheless, for the airfoil flow, with a sharp trailing-edge, residuals were fast converging for a subcritical flow on a sequence of grids. For supercritical flow, residuals converged slower on some intermediate grids than on the finest grid or the two coarsest grids.

Recent Advances in Agglomerated Multigrid

“…As mentioned above, a linear multigrid scheme (as in Ref. [20]) would be beneficial in reducing this variability and efficiently providing convergence well beyond this threshold.…”

“…This preconditioner has scalable parallel performance and used previously as a smoother for multigrid. 13,14 The properties of this defect-correction method as a solver and a smoother for multigrid have been studied extensively 19,20 . As a further development, we envision that linear multigrid would be beneficial within the preconditioner stage of the current approach.…”

“…Lucas et al 17 use divided-difference Jacobians and apply some techniques to reduce bandwidth, including lumping and approximations to the viscous terms. We use a preconditioner based on a defect-correction Jacobian approximation 19,20 with an adaptive time step control. The preconditioner uses the linearization of a first-order approximation for inviscid fluxes; the linearization of viscous fluxes is exact on primal grids and approximated on the agglomerated grids.…”

“…We use a preconditioner based on a defect-correction Jacobian approximation 19,20 with an adaptive time step control. The preconditioner uses the linearization of a first-order approximation for inviscid fluxes; the linearization of viscous fluxes is exact on primal grids and approximated on the agglomerated grids.…”