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A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver

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Abstract

In this paper, we improve the Navier–Stokes flow solver developed by Sun et al. based on the spectral volume method (SV) 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 is adopted here. In the original SV solver, the viscous flux was computed with a local discontinuous Galerkin (LDG) type approach. In this study, an interior penalty approach is developed and tested for both the Laplace and Navier–Stokes equations. In addition, the second method of Bassi and Rebay (also known as BR2 approach) is also implemented in the SV context, and also tested. Their convergence properties are studied with the implicit BLU-SGS approach. Fourier analysis revealed some interesting advantages for the penalty method over the LDG method. A convergence speedup of up to 2-3 orders is obtained with the implicit method. The convergence was further enhanced by employing a p-multigrid algorithm. Numerical simulations were performed using all the three viscous flux formulations and were compared with existing high order simulations (or in some cases, analytical solutions). The penalty and the BR2 approaches displayed higher accuracy than the LDG approach. In general, the numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.

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References

  1. Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comput. Phys. 114, 45–58 (1994). doi:10.1006/jcph.1994.1148

    Article  MathSciNet  MATH  Google Scholar 

  2. Aftosmis, M., Gaitonde, D., Tavares, T.S.: Behavior of linear reconstruction techniques on unstructured meshes. AIAA J. 33, 2038–2049 (1995). doi:10.2514/3.12945

    Article  MATH  Google Scholar 

  3. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982). doi:10.1137/0719052

    Article  MathSciNet  MATH  Google Scholar 

  4. Barth, T.J., Frederickson, P.O.: High-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper No. 90-0013 (1990)

  5. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier Stokes equations. J. Comput. Phys. 131, 267–279 (1997). doi:10.1006/jcph.1996.5572

    Article  MathSciNet  MATH  Google Scholar 

  6. Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2d Euler equations. J. Comput. Phys. 138, 251–285 (1997). doi:10.1006/jcph.1997.5454

    Article  MathSciNet  MATH  Google Scholar 

  7. Bassi, F., Rebay, S.: GMRES discontinuous Galerkin solution of the compressible Navier–Stokes equations. In: Karniadakis, G.E., Cockburn, B., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 197–208. Springer, Berlin (2000)

    Google Scholar 

  8. Bassi, F., Rebay, S.: Numerical solution of the Euler equations with a multiorder discontinuous Finite element method. In: Proceedings of the Second International Conference on Computational Fluid Dynamics, Sydney, Australia, 15–19 July 2002

  9. Baumann, C.E.: An hp-adaptive discontinuous finite element method for computational fluid dynamics. Ph.D. dissertation, University of Texas at Austin, December 1997

  10. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16, 365–378 (2000). doi:10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, R.F., Wang, Z.J.: Fast, block lower-upper symmetric Gauss Seidel scheme for arbitrary grids. AIAA J. 38(12), 2238–2245 (2000)

    Article  Google Scholar 

  12. Chen, Q.Y.: Partitions of a simplex leading to accurate spectral (finite) volume reconstruction. SIAM J. Sci. Comput. 27(4), 1458–1470 (2006). doi:10.1137/030601387

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, Q.Y.: Partitions for spectral finite volume reconstruction in the tetrahedron. J. Sci. Comput. 29(3), 299–319 (2006). doi:10.1007/s10915-005-9009-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection diffusion system. SIAM J. Numer. Anal. 35, 2440–2463 (1998). doi:10.1137/S0036142997316712

    Article  MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001). doi:10.1023/A:1012873910884

    Article  MathSciNet  MATH  Google Scholar 

  16. Delanaye, M., Liu, Y.: Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CP (1999)

  17. Fidkowski, K.J., Oliver, T.A., Lu, J., Darmofal, D.L.: p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 207, 92–113 (2005). doi:10.1016/j.jcp.2005.01.005

    Article  MATH  Google Scholar 

  18. Gottlieb, S.: On high-order strong stability preserving Runge–Kutta and multi step time discretizations. J. Sci. Comput. 25(1/2), 105–128 (2005)

    MathSciNet  Google Scholar 

  19. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67, 73–85 (1998). doi:10.1090/S0025-5718-98-00913-2

    Article  MathSciNet  MATH  Google Scholar 

  20. Haga, T., Ohnishi, N., Sawada, K., Masunaga, A.: Spectral volume computation of flowfield in aerospace application using Earth Simulator. AIAA Paper, 2006–2823

  21. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231 (1987). doi:10.1016/0021-9991(87)90031-3

    Article  MathSciNet  MATH  Google Scholar 

  22. Helenbrook, B.T., Atkins, H.L.: Application of p-multigrid to discontinuous Galerkin formulations of the Poisson equation. AIAA J. 44, 566–575 (2006). doi:10.2514/1.15497

    Article  Google Scholar 

  23. Jameson, A., Yoon, S.: Lower–upper implicit schemes with multiples grids for the Euler equations. AIAA J. 25(7), 929–935 (1987). doi:10.2514/3.9724

    Article  Google Scholar 

  24. Liang, C., Kannan, R., Wang, Z.J.: A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured grids. Comput. Fluids 38(2), 254–265 (2009)

    Article  Google Scholar 

  25. Liou, M.-S., Steffen, C.: A New Flux Splitting Scheme. J. Comput. Phys. 107, 23–39 (1993). doi:10.1006/jcph.1993.1122

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, Y., Vinokur, M., Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys. 212, 454–472 (2006). doi:10.1016/j.jcp.2005.06.024

    Article  MathSciNet  MATH  Google Scholar 

  27. Luo, H., Baum, J.D., Löhner, R.: A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys. 211, 767–783 (2006). doi:10.1016/j.jcp.2005.06.019

    Article  MathSciNet  MATH  Google Scholar 

  28. Maday, Y., Munoz, R.: Spectral element multigrid, Part 2: Theoretical justification. Tech. Rep. 88-73, ICASE (1988)

  29. Mavriplis, D.J.: Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. J. Comput. Phys. 145, 141–165 (1998). doi:10.1006/jcph.1998.6036

    Article  MathSciNet  MATH  Google Scholar 

  30. Mavriplis, D.J., Jameson, A., Martinelli, L.: Multigrid solution of the Navier–Stokes equations on triangular meshes. AIAA Paper 89–0120 (1989)

  31. Nastase, C.R., Mavriplis, D.J.: High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys. 213, 330–357 (2006). doi:10.1016/j.jcp.2005.08.022

    Article  MATH  Google Scholar 

  32. Radespiel, R., Swanson, R.C.: An investigation of cell-centered and cell vertex multigrid schemes for Navier–Stokes equations. AIAA Paper No. 89-0543 (1989)

  33. Rasetarinera, P., Hussaini, M.Y.: An efficient implicit discontinuous spectral Galerkin method. J. Comput. Phys. 172, 718–738 (2001). doi:10.1006/jcph.2001.6853

    Article  MATH  Google Scholar 

  34. Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981). doi:10.1016/0021-9991(81)90128-5

    Article  MathSciNet  MATH  Google Scholar 

  35. Ronquist, E.M., Patera, A.T.: Spectral element multigrid. I. Formulation and numerical results. J. Sci. Comput. 2(4), 389–406 (1987). doi:10.1007/BF01061297

    MathSciNet  Google Scholar 

  36. Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)

    MathSciNet  Google Scholar 

  37. Saad, Y., Schultz, M.H.: GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 865 (1986)

    MathSciNet  Google Scholar 

  38. Sharov, D., Nakahashi, K.: Low speed preconditioning and LUSGS scheme for 3D viscous low computations on unstructured grids. AIAA Paper No. 98–0614, January 1998

  39. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MATH  Google Scholar 

  40. Shu, C.-W.: Navier–Stokes equations on triangular meshes. AIAA J. (1988). doi:10.1137/0909073

    Google Scholar 

  41. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988). doi:10.1016/0021-9991(88)90177-5

    Article  MathSciNet  MATH  Google Scholar 

  42. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability preserving time discretization methods

  43. Sun, Y., Wang, Z.J.: Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method. Commun. Comput. Phys. (submitted)

  44. Sun, Y., Wang, Z.J.: Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grid. Int. J. Numer. Methods Fluids 45(8), 819–838 (2004). doi:10.1002/fld.726

    Article  MATH  Google Scholar 

  45. Sun, Y., Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids VI: Extension to viscous flow. J. Comput. Phys. 215, 41–58 (2006). doi:10.1016/j.jcp.2005.10.019

    Article  MathSciNet  MATH  Google Scholar 

  46. Van Den Abeele, K.V., Broeckhoven, T., Lacor, C.: Dispersion and dissipation properties of the 1d spectral volume method and application to a p-multigrid algorithm. J. Comput. Phys. (2007, in press)

  47. Van Den Abeele, K.V., Lacor, C.: An accuracy and stability study of the 2D spectral volume method. J. Comput. Phys. 226(1), 1007–1026 (2007). doi:10.1016/j.jcp.2007.05.004

    Article  MathSciNet  MATH  Google Scholar 

  48. Van Leer, B.: Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361 (1974). doi:10.1016/0021-9991(74)90019-9

    Article  Google Scholar 

  49. Van Leer, B.: Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101 (1979). doi:10.1016/0021-9991(79)90145-1

    Article  Google Scholar 

  50. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178, 210 (2002). doi:10.1006/jcph.2002.7041

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys. 179, 665 (2002). doi:10.1006/jcph.2002.7082

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids III: extension to one-dimensional systems. J. Sci. Comput. 20, 137 (2004). doi:10.1023/A:1025896119548

    Article  MathSciNet  MATH  Google Scholar 

  53. Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations. J. Comput. Phys. 194, 716 (2004). doi:10.1016/j.jcp.2003.09.012

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, Z.J., Liu, Y.: Extension of the spectral volume method to high-order boundary representation. J. Comput. Phys. 211, 154–178 (2006). doi:10.1016/j.jcp.2005.05.022

    Article  MATH  Google Scholar 

  55. Zhang, M., Shu, C.W.: An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13, 395–413 (2003). doi:10.1142/S0218202503002568

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, M., Shu, C.W.: An analysis and a comparison between the discontinuous Galerkin method and the spectral finite volume methods. Comput. Fluids 34(4–5), 581–592 (2005). doi:10.1016/j.compfluid.2003.05.006

    Article  MATH  Google Scholar 

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  1. Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, IA, 50011, USA

    R. Kannan & Z. J. Wang

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Kannan, R., Wang, Z.J. A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver. J Sci Comput 41, 165 (2009). https://doi.org/10.1007/s10915-009-9269-1

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