Abstract
The methods most frequently used in computational fluid mechanics for solving the compressible Navier-Stokes or compressible Euler equations are finite volume schemes on structured or on unstructured grids. First order as well as higher order space discretizations of MUSCL type, including flux limiters and higher order Runge- Kutta methods for the time discretization, guarantee robust and accurate schemes. But there is an important difficulty. If one increases the order, the stencil for the space discretization increases too, and the scheme becomes very expensive. Therefore schemes with more compact stencils are necessary. Discontinuous Galerkin schemes in the sense of [3] are of this type. They are identical to finite volume schemes in the case of formal first order, and for higher order they use nonconformal ansatz functions whose restrictions to single cells are polynomials of higher order. Therefore they seem to be more efficient and it is of highest interest to compare finite volume and discontinuous Galerkin methods for real applications with respect to their efficiency. Experiences [1] with the Euler equations of gas dynamics indicate that the discontinuous Galerkin methods have some advantages. Since there are no systematic studies available in the literature, we will present in this paper some numerical experiments for hyperbolic conservation laws in multiple space dimensions to compare their efficiency for different situations. As important instances of hyperbolic conservation laws we consider the Euler equations of gas dynamics and Lundquist’s equations of ideal magneto-hydrodynamics (MHD). Furthermore we have found a new limiter which improves the results from [14]. Similar studies have been done in [4].
Keywords
- Finite Volume
- Finite Volume Method
- Unstructured Grid
- Discontinuous Galerkin Method
- Posteriori Error Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Becker, J.: Entwicklung eines effizienten Verfahrens zur Lösung hyperbolischer Differentialgleichungen. Universität Freiburg, Dissertation (1999), http://www.freidok.uni-freiburg.de/volltexte/123/
Cockburn, B., Hou, S., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp. 54 (1990), no. 190, 545–581
Cockburn, B., Karniadakis, E., Shu, C.-W.: The development of discontinuous Galerkin methods. Lecture Notes in Computational Science and Engineering 11 (2000), 3–52
Dolejsi, V, Feistauer, M., Schwab, C: On some aspects of the discontinuous Galerkin finite element method for conservation laws. To appear in: Mathematics and Computers in Simulation
Friedrichs, K.O.: On the laws of relativistic electro-magneto-fluid dynamics. Comm. pure appl. Math. 27 (1974), 749–808
Kröner, D., Noelle, S., Rokyta, M.: Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. Numer. Math., 71 (1995), no. 4, 527–560
Kröner, D.: Numerical schemes for conservation laws. Wiley-Teubner series advances in numerical mathematics. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, first edition (1997)
Kröner, D., Ohlberger, M.: A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69 (2000), no. 229, 25–39
Küther, M.: Error estimates for second order finite volume schemes using a TVD-Runge-Kutta time discretization for a nonlinear scalar hyperbolic conservation law. East-West J. Numer. Math. 8 (2000), no. 4, 299–322
Schnitzer, T.: Discontinuous Galerkin Verfahren angewandt auf die MHD-Gleichungen, Diplomarbeit (2002)
Wesenberg, M.: Finite-Volumen-Verfahren für die Gleichungen der Magnetohydrodynamik in ein und zwei Raumdimensionen, Diplomarbeit (1998)
Wesenberg, M.: Efficient MHD Riemann solvers for simulations on unstructured triangular grids. To appear in: J. of Numer. Math.
Wesenberg, M.: Efficient higher-order finite volume schemes for (real gas) magnetohydrodynamics. PhD thesis (2002)
Wierse, M.: A new theoretically motivated higher order upwind scheme on unstructured grids of simplices. Adv. Comput. Math. 7 (1997), no. 3, 303–335
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Dedner, A., Kröner, D., Rohde, C., Schnitzer, T., Wesenberg, M. (2003). Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_30
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DOI: https://doi.org/10.1007/978-3-642-55627-2_30
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