Abstract
In this paper the recently developed space-time expansion discontinuous Galerkin (STE-DG) approach for the two dimensional unsteady compressible Navier-Stokes equations is presented. The basis of the scheme is a weak formulation of the Navier-Stokes equations, where special care of the second order terms is taken. The spatial polynomial of the DG approach is expanded in time using the so called Cauchy-Kovalevskaya (CK) procedure. With a polynomial of order N in space the CK procedure generates an approximation of order N in time as well yielding a scheme of order N+1 in space and time. The locality and the space-time nature of the presented method give the interesting feature that the time steps may be different in each grid cell. Hence, we drop the common global time steps and propose for a time-accurate solution that any grid cell runs with its own time step determined by the local stability restriction. In spite of the local time steps the scheme is conservative, fully explicit, and as in the DG approach the polynomial order could be chosen arbitrarily, the scheme is theoretically of arbitrary order of accuracy in space and time for transient calculations.
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References
F. Bassi and S. Rebay: ”A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations”. Journal of Computational Physics 131, 1997, pp. 267–279.
M. Dumbser: ”Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains”. Shaker Verlag, Aachen, 2005
M. Dumbser, C.-D. Munz: ”Arbitrary High Order discontinuous Galerkin schemes”. Numerical Methods for Hyperbolic and Kinetic Problems, IRMA Series in Mathematics and Theoretical Physics, EMS Publishing House, 2005, pp. 295–333.
G. Gassner, F. Lörcher, C.-D. Munz:”A Contribution to the Construction of Diffusion Fluxes for Finite Volume and Discontinuous Galerkin Schemes”, J. Comput. Phys., 2006, doi:10.1016/j.jcp.2006.11.004
G. Gassner, F. Lörcher, C.-D. Munz:”A discontinuous Galerkin scheme based on a spacetime expansion. II. Viscous flow equations in multi dimensions”, submitted to J. Sci. Comp., 2006
A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy: ”Uniformly High Order Accurate Essentially Non-oscillatory Schemes III”. J. Comput. Phys. 71, 1987, pp. 231–303.
O. Inoue, N. Hatakeyama: ”Sound Generation by a Two-Dimensional Circular Cylinder in a Uniform How”. Journal of Fluid Mechanics 471, 2002, pp. 285–314.
F. Lörcher, G. Gassner, C.-D. Munz: ”A discontinuous Galerkin scheme based on a space-time expansion. I. Inviscid compressible flow in one space dimension”, J. Sci. Comp., 2007, doi:10.1007/sl0915-007-9128-x
E.F Toro: ”Riemann Solvers and Numerical Methods for FluidDynamics”. Springer, 1997.
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Lörcher, F., Gassner, G., Münz, CD. (2007). The Space-Time Expansion DG Method. In: Tropea, C., Jakirlic, S., Heinemann, HJ., Henke, R., Hönlinger, H. (eds) New Results in Numerical and Experimental Fluid Mechanics VI. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol 96. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74460-3_19
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DOI: https://doi.org/10.1007/978-3-540-74460-3_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74458-0
Online ISBN: 978-3-540-74460-3
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