Abstract
We propose an explicit in time- discontinuous Galerkin scheme on moving grids using the arbitrary Lagrangian–Eulerian approach for one-dimensional Euler equations. The grid is moved with a velocity that is close to the local fluid velocity, which considerably reduces the numerical dissipation in the Riemann solvers. Local grid refinement and coarsening are performed to maintain the mesh quality and avoid very small or large cells. Second-, third-, and fourth-order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.
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References
J. Donea, A. Huerta, J.-P. Ponthot, A. Rodríguez-Ferran, Arbitrary Lagrangian–Eulerian Methods (Wiley, New York, 2004)
E. Johnsen, J. Larsson, A.V. Bhagatwala, W.H. Cabot, P. Moin, B.J. Olson, P.S. Rawat, S.K. Shankar, B. Sjögreen, H.C. Yee, X. Zhong, S.K. Lele, Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229(4), 1213–1237 (2010)
H. Luo, J.D. Baum, R. Löhner, On the computation of multi-material flows using ALE formulation. J. Comput. Phys. 194(1), 304–328 (2004)
B. Owren, M. Zennaro, Derivation of efficient, continuous, explicit runge-kutta methods. SIAM J. Sci. Stat. Comput. 13(6), 1488–1501 (1992)
V. Springel, E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh. Mon. Not. R. Astron. Soc. 401(2), 791–851 (2010)
H. Weizhang, R.D. Russell, Adaptive Moving Mesh Methods, Applied Mathematical Sciences (Springer, Berlin, 2011)
X. Zhang, C.-W. Shu, On positivity-preserving high order discontinuous galerkin schemes for compressible euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918–8934 (2010)
Acknowledgements
The authors gratefully acknowledge the financial support received from the Airbus Foundation Chair on Mathematics of Complex Systems established in TIFR-CAM, Bangalore, for carrying out this work.
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Badwaik, J., Chandrashekar, P. (2018). Arbitrary Lagrangian–Eulerian Discontinuous Galerkin Method for 1D Euler Equations. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_26
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DOI: https://doi.org/10.1007/978-3-319-91545-6_26
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