×

zbMATH — the first resource for mathematics

The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions. (English) Zbl 1391.76347
Summary: This paper presents a new cell-centered Lagrangian scheme for two-dimensional compressible flow. The new scheme uses a semi-Lagrangian form of the Euler equations. The system of equations is discretized by discontinuous Galerkin (DG) method using the Taylor basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently by a nodal solver. The mesh moves with the fluid flow. The time marching is implemented by a class of the Runge-Kutta (RK) methods. A WENO reconstruction is used as a limiter for the RKDG method. The scheme is conservative for the mass, momentum and total energy, and obeys the geometrical conservation law. The scheme maintains high-order accuracy and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations
Software:
CAVEAT
PDF BibTeX Cite
Full Text: DOI
References:
[1] Caramana, E. J.; Burton, D. E.; Shashkov, M. J., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J Comput Phys, 146, 227-262, (1998) · Zbl 0931.76080
[2] Von Neumann, J.; Richtmyer, R. D., A method for the numerical calculations of hydrodynamical shocks, J Appl phys, 21, 232-238, (1950) · Zbl 0037.12002
[3] Campbell, J. C.; Shashkov, M. J., A tensor artificial viscosity using a mimetic finite difference algorithm, J Comput Phys, 172, 739-765, (2001) · Zbl 1002.76082
[4] Cheng, J.; Shu, C.-W., A high-order ENO conservative Lagrangian type scheme for the compressible Euler equations, J Comput Phys, 227, 1567-1596, (2007) · Zbl 1126.76035
[5] Cheng, J.; Shu, C.-W., A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, Commun Comput Phys, 4, 1008-1024, (2008) · Zbl 1364.76111
[6] Maire, P.-H.; Abgrall, R.; Breil, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J Sci Comput, 29, 1781-1824, (2007) · Zbl 1251.76028
[7] Maire, P.-H.; Breil, J., A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems, Int J Numer Meth Fluids, 56, 1417-1423, (2007) · Zbl 1151.76021
[8] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J Comput Phys, 228, 2391-2425, (2009) · Zbl 1156.76434
[9] Hui, W. H.; Li, P. Y.; Li, Z. W., A unified coordinate system for solving the two-dimensional Euler equations, J Comput Phys, 153, 596-637, (1999) · Zbl 0969.76061
[10] Jia, Z. P.; Zhang, S. D., A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two dimensions, J Comput Phys, 230, 2496-2522, (2011) · Zbl 1316.76049
[11] Loubère, R.; Ovadia, J.; Abgrall, R., A Lagrangian discontinuous Galerkin-type method on unstructed meshes to solve hydrodynamics problems, Int J Numer Meth Fluids, 44, 645-663, (2004) · Zbl 1067.76591
[12] G.Z. Zhao. Variational iteration method And RKDG finite element method used in Lagrangian coordinate, Ph D Thesis. China Academy of Engineering Physics; 2011.
[13] Luo, H.; Baum, J. D.; Löhner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J Comput Phys, 227, 8875-8893, (2008) · Zbl 1391.76350
[14] Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun Comput Phys, 14, 1174-1206, (2013) · Zbl 1388.65075
[15] Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems:applications to compressible multi-phase flows, Comput Fluids, 86, 405-432, (2013) · Zbl 1290.76081
[16] Vilar, F.; Maire, P.-H.; Abgrall, R., Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics, Comput Fluids, 46, 498-504, (2011) · Zbl 1433.76093
[17] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag Berlin · Zbl 0923.76004
[18] Dukowicz, J. K.; Meltz, B. J.A., Vorticity errors in multi-dimensional Lagrangian codes, J Comput Phys, 99, 115-134, (1992) · Zbl 0743.76058
[19] Després, B.; Mazeran, C., Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Arch Ration Mech Anal, 178, 327-372, (2005) · Zbl 1096.76046
[20] Qiu, J.; Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J Sci Comput, 26, 907-929, (2005) · Zbl 1077.65109
[21] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J Comput Phys, 141, 199-224, (1998) · Zbl 0920.65059
[22] Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: two-dimensional case, Comput Fluids, 34, 642-663, (2005) · Zbl 1134.65358
[23] Wilkins, M. L., Calculation of elastic-plastic flow, (Methods in computational physics, vol. 3, (1964), Academic Press New York), 211-263
[24] Vilar, F., Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics, Comput Fluids, 64, 64-73, (2012) · Zbl 1365.76129
[25] Zhu, J.; Qiu, J.; Shu, C.-W.; Dumbser, M., Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes, J Comput Phys, 227, 9, 4330-4353, (2008) · Zbl 1157.65453
[26] Maire, P.-H., A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Int J Numer Meth Fluids, 65, 1281-1294, (2011) · Zbl 1429.76089
[27] Addessio FL, Carroll DE, Dukowicz JK, Harlow FH, Johnson JN, Kashiwa BA, et al. CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip. Los Alamos National Laboratory, UC-32; 1988.
[28] Maire, P.-H., A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput Fluids, 46, 341-347, (2011) · Zbl 1433.76137
[29] Landau, L.; Lifchitz, E., Mécanique des fluides, (1989), Mir Moscow
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.