On the solution of linear algebraic systems arising from the semi–implicit DGFE discretization of the compressible Navier–Stokes equations

Dolejší, Vít

Kybernetika, Tome 46 (2010) no. 2, pp. 260-280

Abstract (VO)

We deal with the numerical simulation of a motion of viscous compressible fluids. We discretize the governing Navier–Stokes equations by the backward difference formula – discontinuous Galerkin finite element (BDF-DGFE) method, which exhibits a sufficiently stable, efficient and accurate numerical scheme. The BDF-DGFE method requires a solution of one linear algebra system at each time step. In this paper, we deal with these linear algebra systems with the aid of an iterative solver. We discuss the choice of the preconditioner, stopping criterion and the choice of the time step and propose a new strategy which leads to an efficient and accurate numerical scheme.

Détail

Classification : 35Q35, 65L06, 65M22, 76M10, 76N15, 76N99
Keywords: discontinuous Galerkin method; compressible Navier–Stokes equations; linear algebra problems; preconditioning; stopping criterion; choice of the time step

Bibliographie

[1] Arnold, D. N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982), 4, 742–760. | DOI | MR | Zbl

[2] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 5, 1749–1779. | DOI | MR | Zbl

[3] 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 (1997), 267–279. | DOI | MR | Zbl

[4] Bassi, F., Rebay, S.: A high order discontinuous Galerkin method for compressible turbulent flow. In: Discontinuous Galerkin Method: Theory, Computations and Applications (B. Cockburn, G. E. Karniadakis, and C. W. Shu, eds.), (Lecture Notes in Computat. Sci. Engrg. 11.) Springer-Verlag, Berlin 2000, pp. 113–123. | MR

[5] Baumann, C. E., Oden, J. T.: A discontinuous $hp$ finite element method for the Euler and Navier-Stokes equations. Internat. J. Numer. Methods Fluids 31 (1999), 1, 79–95. | DOI | MR | Zbl

[6] Ciarlet, P. G.: The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam – New York – Oxford 1979. | MR

[7] Cockburn, B., Hou, S., Shu, C. W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element for conservation laws IV: The multi-dimensional case. Math. Comp. 54 (1990), 545–581. | MR

[8] Dawson, C. N., Sun, S., Wheeler, M. F.: Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Engrg. 193 (2004), 2565–2580. | DOI | MR | Zbl

[9] Dolejší, V.: On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations. Internat. J. Numer. Methods Fluids 45 (2004), 1083–1106. | DOI | MR

[10] Dolejší, V.: Semi-implicit interior penalty discontinuous Galerkin methods for viscous compressible flows. Commun. Comput. Phys. 4 (2008), 2, 231–274. | MR

[11] Dolejší, V., Kůs, P.: Adaptive backward difference formula – discontinuous Galerkin finite element method for the solution of conservation laws. Internat. J. Numer. Methods Engrg. 73 (2008), 12, 1739–1766. | DOI | MR

[12] Dolejší, V.: Discontinuous Galerkin method for the numerical simulation of unsteady compressible flow. WSEAS Trans. on Systems 5 (2006), 5, 1083–1090.

[13] Dolejší, V., Feistauer, M.: Semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. J. Comput. Phys. 198 (2004), 2, 727–746. | DOI | MR

[14] Dumbser, M., Munz, C. D.: Building blocks for arbitrary high-order discontinuous Galerkin methods. J. Sci. Comput. 27 (2006), 215–230. | DOI | MR

[15] Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Oxford University Press, Oxford 2003. | MR

[16] Feistauer, M., Kučera, V.: On a robust discontinuous Galerkin technique for the solution of compressible flow. J. Comput. Phys. 224 (2007), 1, 208–221. | DOI | MR

[17] Feistauer, M., Kučera, V., Prokopová, J.: Discontinuous Galerkin solution of compressible flow in time dependent domains. Math. Comput. Simulations 80 (2010), 8, 1612-1623. | DOI | MR

[18] Hairer, E., Norsett, S. P., Wanner, G.: Solving ordinary differential equations I, Nonstiff problems. (Springer Series in Computational Mathematics No. 8.) Springer Verlag, Berlin 2000. | MR

[19] Hartmann, R., Houston, P.: Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation. Internat. J. Numer. Anal. Model. 1 (2006), 1–20. | MR | Zbl

[20] Klaij, C. M., Vegt, J. van der, Ven, H. V. der: Pseudo-time stepping for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 219 (2006), 2, 622–643. | DOI | MR

[21] Lörcher, F., Gassner, G., Munz, C. D.: A discontinuous Galerkin scheme based on a spacetime expansion. I. Inviscid compressible flow in one space dimension. J. Sci. Comput. 32 (2007), 2, 175–199. | DOI | MR

[22] Rivière, B., Wheeler, M. F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3 (1999), 3-4, 337–360. | MR

[23] Watkins, D. S.: Fundamentals of Matrix Computations. (Pure and Applied Mathematics, Wiley-Interscience Series of Texts, Monographs, and Tracts.) John Wiley , New York 2002. | MR