A Study of Viscous Flux Formulations for an Implicit P-Multigrid Spectral Volume Navier Stokes Solver

In this paper, we improve the Navier-Stokes flow solver developed by Sun et al based on the spectral volume method in the following two aspects: the development of a more efficient implicit/p-multigrid solution approach, and the use of a new viscous flux formula. An implicit preconditioned LU-SGS p-multigrid method developed for the spectral difference (SD) Euler solver by Liang et al is adopted here. In the original SV solver, the viscous flux was computed with a local discontinuous Galerkin-type approach. In this study, a penalty approach based on the first method of Bassi and Rebay is suggested and tested for both the Laplace and Navier-Stokes equations. The second method of Bassi and Rebay is also tested for the Laplace equation. Their convergence properties are studied in the context of the BLU-SGS approach. Fourier analysis revealed some interesting advantages for the penalty method over the LDG method. A convergence speedup of up to 2 orders is obtained with the implicit method. The convergence was further enhanced by employing a p-multigrid algorithm. The numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.