Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders

François Vilar (1) , Chi-Wang Shu (1) , P.H. Maire (2)

1 Department of Applied Mathematics
2 CESTA - Centre d'études scientifiques et techniques d'Aquitaine

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in this Lagrangian gas dynamics framework. To this end, we first focus on the one-dimensional case. After briefly recalling how to derive Lagrangian forms of the gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS. It enables us to derive time step conditions ensuring the desired positivity property, as well as L 1 stability of the specific volume and total energy over the domain. Then, making use of the work presented in [74, 75, 15], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. This whole analysis is finally applied to the two-dimensional case, and shown to fit a wide range of numerical schemes in the literature, such as the GLACE scheme [12], the EUCCLHYD scheme [55], the GLACE scheme on conical meshes [8], and the LCCDG method [72]. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. Finally, let us emphasize that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes.

Mots clés: positivity-preserving high-order methods; cell-centered Lagrangian schemes; updated and total Lagrangian formulations; Godunov-type method; unstructured moving grid; multi-material compressible flows; equations of state; Riemann solver

Domaines: Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]

François Vilar, Chi-Wang Shu, P.H. Maire. Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. 2015. ⟨hal-01199605⟩