Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

• Published in: Journal of computational physics Vol. 229; no. 3; pp. 828 - 850
• Main Authors: Parsani, M., Van den Abeele, K., Lacor, C., Turkel, E.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.02.2010; Elsevier
• Subjects: Algorithms; Boundary element method; Computational techniques; Cylinders; Exact sciences and technology; High-order methods; Implicit LU-SGS algorithm; Laminar; Mach number; Mathematical analysis; Mathematical methods in physics; Navier-Stokes equations; Navier–Stokes; p-Multigrid; Physics; Runge-Kutta method; Von Neumann analysis; Navier–Stokes; p-Multigrid; High-order methods; Implicit LU-SGS algorithm; Von Neumann analysis; Damping; Spectral method; Fluid dynamics; Aspect ratio; Euler scheme; Calculation methods; Advection; Algorithms; Laminar flow; Multigrid; Convergence rate; Navier-Stokes; Calculation; Cylinders; Navier-Stokes equations; Mach number
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2009.10.014

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.