Multigrid Method for the Euler and Navier-Stokes Equations

Abstract

Results are presented of a study to implement convergence acceleration techniques based on the multigrid concept in a three-dimensional computer code developed for solving Euler and Navier-Stokes equations. The multigrid method implemented is the full approximation storage-full multigrid algorithm, which is applicable to nonlinear equations. Multigrid convergence estimates based on a single-grid or bigrid stability analysis are performed. The latter was found to yield a better prediction of practical multigrid convergence rates. In typical applications, for both laminar and turbulent ows, savings of up to 50% in CPU time were obtained with the multigrid implementation. Nomenclatureviscous ux vectors e = error vector e 0 = total speci c internal energy g = computational grid in the multigrid sequence I = 1 I H h = restriction operator I h H = interpolation operator I = identity matrix J = transformation Jacobian K = coarse grid correction matrix K L N = Fourier symbols k = turbulent kinetic energy L 1 L 2 L 3 = alternating direction implicit operators M = bigrid ampli cation matrix Ma = Mach number P = forcing term in multigrid procedure p = pressure Q = solution vector R = residual vector Re = Reynolds number R 0 R 1 R 2 S 0 S 1 S 2 = viscous ux Jacobians Y 0 Y 1 Y 2 S 1 S 2 = relaxation or smoothing operator t = time U 0 = constant amplitude vector u = velocity components x y z = Cartesian coordinates = speci c heat capacity ratio = incremental change x y z = nite difference operators = dissipation rate of turbulent kinetic energy i e = coef cients of implicit and explicit arti cial dissipation x y z = Fourier modes in x, y, and z direction = ampli cation factor bg = bigrid ampli cation factor sg = smoothing factor Presented as Paper 96t k t = eddy and thermal viscosity 1 2 = pre-and postrelaxation counters = computational coordinate system = density mg = multigrid convergence factor Subscripts and Superscriptsh H = ne and coarse grid levels n n 1 n 2 = time step or iteration number