×

A high order cell centred dual grid Lagrangian Godunov scheme. (English) Zbl 1290.76078

Summary: A first order cell centred Lagrangian Godunov scheme based upon the use of a dual grid to determine vertex velocities was presented by the author and P. L. Roe, Comput. Fluids 46, No. 1, 133–136 (2011; Zbl 1431.76006)]. A second order version of the scheme is presented and results obtained with the new scheme are compared against those obtained with a staggered grid compatible finite element scheme [the author, Int. J. Numer. Methods Fluids 56, No. 8, 953–964 (2008; Zbl 1169.76030)]. The new scheme is shown to provide comparable shock capturing to the staggered grid method while retaining the benefits of reduced mesh imprinting, robustness and improved symmetry preservation observed for the first order cell centred scheme [Zbl 1431.76006]. Two different approaches are also considered for moving the vertices using the dual grid approach, a method which reconstructs nodal velocities at the start of every timestep and a second that carries the nodal velocities as an additional variable.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

Keywords:

Lagrangian; Godunov

Software:

HE-E1GODF; CAVEAT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barlow, A. J.; Roe, P. L., A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput Fluids, 46, 133-136 (2011) · Zbl 1431.76006
[2] Barlow, A. J., A compatible finite element multi-material ALE hydrodynamics algorithm, Int J Numer Methods Fluids, 56, 953-964 (2008) · Zbl 1169.76030
[3] Godunov, S. K., Mater Sb, 47, 271 (1959)
[5] Abgrall, R.; Loubere, R.; Ovadia, J., A Lagrangian discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, Int J Numer Methods Fluids, 44, 645-663 (2004) · Zbl 1067.76591
[6] Despres, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch Rational Mech Anal, 178, 327-372 (2005) · Zbl 1096.76046
[7] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for compressible flow problems, SIAM J Sci Comput, 29, 4, 1781-1824 (2007) · Zbl 1251.76028
[9] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, J Comput Phys, 228, 7, 2391-2425 (2009) · Zbl 1156.76434
[10] Loubère, Rapha’el; Maire, Pierre-Henri; Váchal, Pavel, A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell centered multidimensional approximate Riemann solver, Proc Comput Sci, 1, 1, 1931-1939 (2010)
[11] Maire, P.-H.; Loubère, R.; Váchal, P., Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun Comput Phys, 10, 4, 940-978 (2011), http://dx.doi.org/10.4208/cicp.170310.251110a · Zbl 1373.76138
[12] Dukowicz, J. K., A general non-iterative Riemann solver for Godunovs method, J Comput Phys, 61, 119-137 (1984)
[13] Dukowicz, J. K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J Comput Phys, 99, 115-134 (1992) · Zbl 0743.76058
[14] Toro, Eleuterio F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction (2009), Springer · Zbl 1227.76006
[15] Einfeldt, B.; Munz, C. D.; Roe, P. L.; Sjogreen, B., On Godunov type methods near low densities, J Comput Phys, 92, 273-294 (1991) · Zbl 0709.76102
[17] Noh, W. F., Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J Comput Phys, 72, 78-120 (1987) · Zbl 0619.76091
[18] Sod, G. A., A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws, J Comput Phys, 27, 1-31 (1978) · Zbl 0387.76063
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.