Convergence Acceleration for Solving the Compressible Navier-Stokes Equations

Abstract

A fully implicit operator for implicit smoothing of residuals in the framework of explicit Runge-Kutta time stepping and multigrid acceleration is developed. To avoid memory overhead associated with storing the complete flux Jacobians, the Jacobians are expressed in terms of Mach number to enable economic computation of all flux Jacobians during iteration. The implicit operator allows increasing Courant-Friedrichs-Lewy numbers of the basic explicit scheme to the order of 100, and it properly addresses the stiffness in the discrete equations associated with highly stretched meshes. The proposed method was applied to different cases of viscous, turbulent airfoil flow, and convergence rates ranging from 0.8 to 0.87 were achieved. Comparing the present scheme to well-tuned state-of-the-art methods using common implicit residual smoothing techniques, CPU time is better than halved.