Polynomial least-squares reconstruction for semi-Lagrangian cell-centered hydrodynamic schemes.

*(English)*Zbl 1176.76078Summary: In Inertial Confinement Fusion (ICF) simulation, use of Lagrangian hydrodynamic numerical schemes is a cornerstone. It avoids mixing of materials and allows for symmetry preservation in dimension two. Recently, B. Després and C. Mazeran [Arch. Ration. Mech. Anal. 178, No. 3, 327–372 (2005; Zbl 1096.76046)] and then P.-H. Maire et al. [SIAM J. Sci. Comput. 29, No. 4, 1781–1824 (2007; Zbl 1251.76028)] proposed an interesting alternative to the historical VNR scheme [J. von Neumann and R.D.Richtmyer, J. Appl. Phys. 21, 232–237 (1950; Zbl 0037.12002)]. These two first order schemes are multidimensional generalizations of the Godunov acoustic solver. Alternatively, a WENO Lagrangian scheme was proposed by J. Cheng and Ch.-W. Shu [J. Comput. Phys. 227, No. 2, 1567–1596 (2007; Zbl 1126.76035)]. This scheme suffers from non-preservation of symmetries and its velocity computation can be discussed.

The aim of this work is to evaluate the later scheme on ICF representative test cases and to derive a polynomial reconstruction that preserves symmetries for the three cell-centered scheme. Since this paper focuses on the approximation of Euler equations, considered test cases are purely hydrodynamic and do not illustrate all difficulties encountered in ICF.

We first briefly recall different schemes used for this study. We then explain the Least-Squares ENO reconstruction that we chose for symmetry preservation and describe the limiting strategy. We finally illustrates the presented results by some representative numerical experiments.

The aim of this work is to evaluate the later scheme on ICF representative test cases and to derive a polynomial reconstruction that preserves symmetries for the three cell-centered scheme. Since this paper focuses on the approximation of Euler equations, considered test cases are purely hydrodynamic and do not illustrate all difficulties encountered in ICF.

We first briefly recall different schemes used for this study. We then explain the Least-Squares ENO reconstruction that we chose for symmetry preservation and describe the limiting strategy. We finally illustrates the presented results by some representative numerical experiments.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |