Effects of high-frequency damping on iterative convergence of implicit viscous solver

Abstract This paper discusses effects of high-frequency damping on iterative convergence of an implicit defect-correction solver for viscous problems. The study targets a finite-volume discretization with a one parameter family of damped viscous schemes. The parameter α controls high-frequency damping: zero damping with α = 0 , and larger damping for larger α ( > 0 ) . Convergence rates are predicted for a model diffusion equation by a Fourier analysis over a practical range of α . It is shown that the convergence rate attains its minimum at α = 1 on regular quadrilateral grids, and deteriorates for larger values of α . A similar behavior is observed for regular triangular grids. In both quadrilateral and triangular grids, the solver is predicted to diverge for α smaller than approximately 0.5. Numerical results are shown for the diffusion equation and the Navier–Stokes equations on regular and irregular grids. The study suggests that α = 1 and 4/3 are suitable values for robust and efficient computations, and α = 4 / 3 is recommended for the diffusion equation, which achieves higher-order accuracy on regular quadrilateral grids. Finally, a Jacobian-Free Newton–Krylov solver with the implicit solver (a low-order Jacobian approximately inverted by a multi-color Gauss–Seidel relaxation scheme) used as a variable preconditioner is recommended for practical computations, which provides robust and efficient convergence for a wide range of α .