We deal with a numerical solution of the (nonlinear hyperbolic) system of the Euler equations, which describe a motion of inviscid compressible flow. We present a higher order numerical scheme with respect to the space as well as time coordinates. This scheme is based on the discontinuous Galerkin method for the space semi-discretization and the backward difference formula for the time discretization. We employ a suitable linearization of inviscid fluxes and an explicit extrapolation in nonlinear terms, which preserve a high order of accuracy and lead to a linear algebraic problem at each time step. Moreover, we discuss a use of non-reflecting boundary conditions at inflow/outflow parts of boundary and present a stabilization technique that avoid spurious oscillations of numerical solution in vicinity of shock waves. Finally, two numerical examples of unsteady compressible flow demonstrating an efficiency of the scheme are presented.
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References
Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138, 251–285 (1997)
Cockburn, B., Karniadakis, G.E., (eds.), C.-W.S.: Discontinuous Galerkin methods. In Lecture Notes in Computational Science and Engineering 11. Springer, Berlin (2000)
Dolejší, V.: Higher order semi-implicit discontinuous Galerkin finite element schemes for compressible flow simulation. In Software and Algorithms of Numerical Mathematics ((submitted) 2005).
Dolejší, V., Felcman, J.: Anisotropic mesh adaptation and its application for scalar diffusion equations. Numer. Methods Partial Differ. Equations, 20, 576–608 (2004)
Dolejší, V., Feistauer, M.: Semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. J. Comput. Phys., 198, 727–746 (2004)
Dolejší, V., Feistauer, M., Schwab, C.: On some aspects of the discontinuous Galerkin finite element method for conservation laws. Math. Comput. Simul., 61, 333–346 (2003)
Feistauer, M., Dolejší, V., Kučera, V.: On a semi-implicit discontinuous Galerkin FEM for the nonstationary compressible Euler equations. In: Asukara, F., Aiso, H., Kawashima, S., Matsumura, A., Nishibata, S., Nishihara, K. (eds.) Hyperbolic Probelms: Theory Numerics and Applications (Tenth international conference in Osaka 2004), Yokohama Publishers, Inc., 391–398 (2006)
Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Oxford University Press, Oxford (2003)
Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys., 183, 508–532 (2002)
Jaffre, J., Johnson, C., Szepessy, A.: Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci, 5, 367–286 (1995)
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin (1997)
van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I: General formulation. J. Comput. Phys., 182, 546–585 (2002)
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Dolejší, V. (2008). Higher Order Numerical Schemes for Hyperbolic Systems with an Application in Fluid Dynamics. In: Benzoni-Gavage, S., Serre, D. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75712-2_6
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DOI: https://doi.org/10.1007/978-3-540-75712-2_6
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