Convergence Acceleration for Multistage Time-Stepping Schemes

Abstract

The convergence of a Runge-Kutta (RK) scheme with multigrid is accelerated by preconditioning with a fully implicit operator. With the extended stability of the Runge-Kutta scheme, CFL numbers as high as 1000 could be used. The implicit preconditioner addresses the stiffness in the discrete equations associated with stretched meshes. Numerical dissipation operators (based on the Roe scheme, a matrix formulation, and the CUSP scheme) as well as the number of RK stages are considered in evaluating the RK/implicit scheme. Both the numerical and computational efficiency of the scheme with the different dissipation operators are discussed. The RK/implicit scheme is used to solve the two-dimensional (2-D) and three-dimensional (3-D) compressible, Reynolds-averaged Navier-Stokes equations. In two dimensions, turbulent flows over an airfoil at subsonic and transonic conditions are computed. The effects of mesh cell aspect ratio on convergence are investigated for Reynolds numbers between 5.7 × 10 6 and 100.0 × 10 6 . Results are also obtained for a transonic wing flow. For both 2-D and 3-D problems, it is demonstrated that the computational time of a well-tuned standard RK scheme can be reduced at least a factor of four.

Convergence acceleration of Runge–Kutta schemes for solving the Navier–Stokes equations

“…As shown in Ref. [36] the computed pressure distribution agrees fairly well with the experimental data. The computed and experimental lift and drag coefficients are given in Table 2.…”

“…If we consider all the cases of Table 8, the computational efficiency relative to the standard scheme is increased by factors between 4 and 9. In addition, the present RK5/implicit scheme exhibits better 3-D performance than observed previously [36]. This can be seen by considering the M ∞ = 0.84 case.…”

“…Convergence quantities for other Reynolds numbers are given in Ref. [36]. For higher Re values the convergence rate with the standard scheme exceeds 0.995.…”

Multi-Stage Filtering of Numerical Solutions With an Application to the Hele-Shaw Problem

“…There are widely known methods, such as the Runge [1] and Richardson [2] rules, as well as the methods of Aitken [3], Neville [4], Romberg [5] and Winn [6], which allow estimating the errors and accelerating the convergence of a solution on the basis of knowledge about the character of the dependence of the error in the numerical method on the number of grid points n . This is also the subject of investigation of many other authors [7][8][9][10][11][12][13][14][15][16]. However, the justification for these methods is based on the representation of the residual term as an infinitely small quantity.…”

Algebraic multigrid within defect correction for the linearized Euler equations

“…They employ geometric multigrid (GMG) smoothed with either explicit Runge-Kutta methods, or an implicit scheme with some approximate Jacobian [4][5][6]. Despite the investigation of many variations on this theme over the years [6][7][8][9][10], satisfactory grid-insensitive convergence has not been achieved for the Navier-Stokes equations. For two-dimensional singleelement aerofoil geometries and high-quality conformal structured grids, rapid grid independent convergence is possible by employing either line-relaxation or directional coarsening in the direction normal to solid walls (where boundary layers are present).…”