Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

doi:10.1007/s42967-019-00054-5

Abstract

Abstract: We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fuid velocity modifed by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refnement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

Key words: Discontinuous Galerkin method, Moving meshes, Arbitrary Lagrangian-Eulerian, Euler equations

CLC Number:

65M60
35L04

Jayesh Badwaik, Praveen Chandrashekar, Christian Klingenberg. Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 541-579.

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References

1. Springel, V.:E pur si muove:Galilean-invariant cosmological hydrodynamical simulations on a moving mesh. Mon. Notices R. Astron. Soc. 401(2), 791-851 (2010)
2. Johnsen, E., Larsson, J., Bhagatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., Rawat, P.S., Shankar, S.K., Sjögreen, B., Yee, H.C., Zhong, X., Lele, S.K.:Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229(4), 1213-1237 (2010)
3. Munz, C.D.:On Godunov-type schemes for Lagrangian gas dynamics. SIAM J. Num. Anal. 31(1), 17-42 (1994)
4. Carré, G., Del Pino, S., Després, B., Labourasse, E.:A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. J. Comput. Phys. 228(14), 5160-5183 (2009)
5. Maire, P.-H.:A high-order cell-centered Lagrangian scheme for two-dimensional compressible fuid fows on unstructured meshes. J. Comput. Phys. 228(7), 2391-2425 (2009)
6. Hirt, C., Amsden, A., Cook, J.:An arbitrary Lagrangian-Eulerian computing method for all fow speeds. J. Comput. Phys. 14(3), 227-253 (1974)
7. Donea, J., Huerta, A., Ponthot, J.-P., Rodríguez-Ferran, A.:Arbitrary Lagrangian-Eulerian Methods. Wiley, Oxford (2004)
8. Walter, B.:High order accurate direct arbitrary-Lagrangian-Eulerian ADER-MOOD fnite volume schemes for non-conservative hyperbolic systems with stif source terms. Commun. Comput. Phys. 21(01), 271-312 (2017)
9. Gaburro, E., Castro, M.J., Dumbser, M.:Well-balanced arbitrary-Lagrangian-Eulerian fnite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity. Mon. Notices R. Astron. Soc. 477(2), 2251-2275 (2018). https://doi.org/10.1093/mnras/sty542.21 June
10. Lomtev, I., Kirby, R., Karniadakis, G.:A discontinuous Galerkin ALE method for compressible viscous fows in moving domains. J. Comput. Phys. 155(1), 128-159 (1999)
11. Venkatasubban, C.S.:A new fnite element formulation for ALE (arbitrary Lagrangian Eulerian) compressible fuid mechanics. Int. J. Eng. Sci. 33(12), 1743-1762 (1995)
12. Wang, L., Persson, P.-O.:High-order discontinuous Galerkin simulations on moving domains using ALE formulations and local remeshing and projections. American Institute of Aeronautics and Astronautics (2015)
13. Weizhang, H., Russell, R.D.:Adaptive Moving Mesh Methods. Applied Mathematical Sciences. Springer, Berlin (2011)
14. He, P., Tang, H.:An adaptive moving mesh method for two-dimensional relativistic hydrodynamics. Commun. Comput. Phys. 11(1), 114-146 (2012)
15. Luo, H., Baum, J.D., Löhner, R.:On the computation of multi-material fows using ALE formulation. J. Comput. Phys. 194(1), 304-328 (2004)
16. Boscheri, W., Dumbser, M., Balsara, D.S.:High-order ADER-WENO ALE schemes on unstructured triangular meshes-application of several node solvers to hydrodynamics and magnetohydrodynamics. Int. J. Numer. Meth. Fluids 76, 737-778 (2014). https://doi.org/10.1002/fd.3947
17. Luo, H., Baum, J.D., Löhner, R.:On the computation of multi-material fows using ALE formulation. J. Comput. Phys. 194, 304-328 (2004). https://doi.org/10.1016/j.jcp.2003.09.026
18. Liu, W., Cheng, J., Shu, C.-W.:High order conservative Lagrangian schemes with Lax-Wendrof type time discretization for the compressible Euler equations. J. Comput. Phys. 228(23), 8872-8891 (2009)
19. Boscheri, W., Dumbser, M.:Arbitrary-Lagrangian-Eulerian one-step WENO fnite volume schemes on unstructured triangular meshes. Commun. Comput. Phys. 14(5), 1174-1206 (2013)
20. Boscheri, W., Dumbser, M.:High order accurate direct arbitrary-Lagrangian-Eulerian ADER-WENO fnite volume schemes on moving curvilinear unstructured meshes. Comput. Fluids 136, 48-66 (2016)
21. Boscheri, W., Dumbser, M., Balsara, D.S.:High-order ADER-WENO ALE schemes on unstructured triangular meshes-application of several node solvers to hydrodynamics and magnetohydrodynamics. Int. J. Num. Methods Fluids 76, 737-778 (2014)
22. Klingenberg, C., Schnücke, G., Xia, Y.:Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws:analysis and application in one dimension. Math. Comput. 86, 1203-1232 (2017)
23. Harten, A., Hyman, J.M.:Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50, 253-269 (1983)
24. Gassner, G., Dumbser, M., Hindenlang, F., Munz, C.-D.:Explicit one-step time discretizations for discontinuous Galerkin and fnite volume schemes based on local predictors. J. Comput. Phys. 230(11), 4232-4247 (2011)
25. Owren, B., Zennaro, M.:Derivation of efcient, continuous, explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput. 13(6), 1488-1501 (1992)
26. Cockburn, B., Lin, S.-Y., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws III:One-dimensional systems. J. Comput. Phys. 84, 90-113 (1989)
27. Cockburn, B., Shu, C.-W.:TVB Runge-Kutta local projection discontinuous Galerkin fnite element method for conservation laws II:General framework. Math. Comput. 52(186), 411-435 (1989)
28. Zhang, X., Shu, C.-W.:On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918-8934 (2010)
29. Zhong, X., Shu, C.-W.:A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys. 232(1), 397-415 (2013)
30. Sod, G.:A survey of several fnite diference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1-31 (1978)
31. Shu, C.-W., Osher, S.:Efcient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988)
32. Titarev, V.A., Toro, E.F.:Finite volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201, 238-260 (2014)
33. Toro, E.F.:Riemann Solvers and Numerical Methods for Fluid Dynamics:A Practical Introduction. Springer, Berlin (1999)
34. Woodward, P., Colella, P.:The numerical simulation of two-dimensional fuid fow with strong shocks. J. Comput. Phys. 54(1), 115-173 (1984)
35. Zhu, J., Qiu, J.:A new ffth order fnite diference WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 318, 110-121 (2016)
36. Loubere, R., Shashkov, M.J.:A subcell remapping method on staggered polygonal grids for arbitraryLagrangian-Eulerian methods. J. Comput. Phys. 209(1), 105-138 (2005)
37. Olivier, G., Alauzet, F.:A new changing-topology ALE scheme for moving mesh unsteady simulations. In:49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, p. 474 (2011)
38. Frederic, Alauzet:A changing-topology moving mesh technique for large displacements. Eng. Comput. 30(2), 175-200 (2014)
39. Roe, P.L.:Approximate Riemann solvers, parameter vectors, and diference schemes. J. Comput. Phys. 43(2), 357-372 (1981)

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