An O(Nm2) plane solver for the compressible Navier-Stokes equations

Abstract

A hierarchical multigrid algorithm for e cient steady solutions to the two-dimensional compressible Navier-Stokes equations is developed and demonstrated. The algorithm applies multigrid in twoways: a Full Approximation Scheme FAS for a nonlinear residual equation and a Correction Scheme CS for a linearized defect correction implicit equation. Multigrid analyses which include the e ect of boundary conditions in one direction are used to estimate the convergence rate of the algorithm for a model convection equation. Three alternating-lineimplicit algorithms are compared in terms of eciency. The analyses indicate that full multigrid e ciency is not attained in the general case; the number of cycles to attain convergence is dependent on the mesh density for high-frequency cross-stream variations. However, the dependence is reasonably small and fast convergence is eventually attained for any given frequency with either the FAS or the CS scheme alone. The paper summarizes numerical computations for which convergence has been attained to within truncation error in a few multigrid cycles for both inviscid and viscous ow simulations on highly stretched meshes. Head, Aerodynamic and Acoustic Methods Branch AAMB, AIAA Fellow. y Member AAMB, AIAA Senior Member. z Member AAMB, AIAA Associate Fellow.

An Implicit Multigrid Method for Turbulent Combustion

“…Excellent results have already been achieved for subsonic and transonic flows [11]. The combination of a full approximation storage (FAS) scheme for the nonlinear equations and a linearized defect correction already enables O(N m 2 ) solutions for transonic laminar test cases [12,13], where N is the number of grid points and m the number of equations. Another promising approach uses different coarsening techniques according to the damping properties of different solvers [14].…”

Convergence of defect-correction and multigrid iterations for inviscid flows

“…6,7 The previous scheme used alternating-line Jacobi and/or colored relaxations that do not provide sufficient damping of high-frequency errors in purely inviscid regions of the flow. Analysis methods were not applied to identify this shortcoming and instead other parts of the algorithm were modified to compensate, namely a pseudo-time step limited by a maximum CFL of O(100) was added to the implicit relaxation operator, relaxation subiterations were performed, and dissipation via entropy fixes to all fields was added to the FDS discretization.…”

“…Analysis of DC convergence for two-dimensional (2D) convection has been previously performed in a semidiscrete setting 6,8 in which boundary conditions in one direction are taken into account. A two-level multigrid analysis 6 showed that although the number of cycles to attain convergence was dependent on the mesh density, the dependence was reasonably small and fast asymptotic convergence was eventually attained. A more detailed study of DC alone 8 showed that an asymptotic convergence of about 0.5 per DC iteration is observed in computations.…”

“…A hierarchical full-approximation scheme (FAS) multigrid method 6,7 with a DC-based relaxation scheme, herein referred to as MG-DC, was previously developed and applied in two dimensions, demonstrating fast convergence of residuals for airfoils at compressible and incompressible conditions. Analysis of DC convergence for two-dimensional (2D) convection has been previously performed in a semidiscrete setting 6,8 in which boundary conditions in one direction are taken into account. A two-level multigrid analysis 6 showed that although the number of cycles to attain convergence was dependent on the mesh density, the dependence was reasonably small and fast asymptotic convergence was eventually attained.…”

“…[1][2][3] On the other hand, DC has been used to solve large-scale turbulent application problems for many years [4][5][6] and relatively fast asymptotic convergence of residuals has been observed in many instances. A hierarchical full-approximation scheme (FAS) multigrid method 6,7 with a DC-based relaxation scheme, herein referred to as MG-DC, was previously developed and applied in two dimensions, demonstrating fast convergence of residuals for airfoils at compressible and incompressible conditions. Analysis of DC convergence for two-dimensional (2D) convection has been previously performed in a semidiscrete setting 6,8 in which boundary conditions in one direction are taken into account.…”