Positivity-preserving Lagrangian scheme for multi-material compressible flow

Juan Cheng Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25043

Journal of Computational Physics, 257, 143-168, 2014
Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear instability and crash of the code. This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian methods. In this paper, we construct both first order and uniformly high order accurate conservative Lagrangian schemes which preserve positivity of physically positive variables such as density and internal energy in the simulation of compressible multi-material flows with general equations of state (EOS). We first develop a positivity-preserving approximate Riemann solver for the Lagrangian scheme solving compressible Euler equations with both ideal and non ideal EOS. Then we design a class of high order positivity-preserving and conservative Lagrangian schemes by using the essentially non-oscillatory (ENO) reconstruction, the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter which can be proven to maintain conservation and uniformly high order accuracy and is easy to implement. One-dimensional and two-dimensional numerical tests for the positivity-preserving Lagrangian schemes are provided to demonstrate the effectiveness of these methods.
Lagrangian method; multi-material compressible flow; positivity-preserving; high order accuracy; general equation of state
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  title={Positivity-preserving Lagrangian scheme for multi-material compressible flow},
  author={Juan Cheng, and Chi-Wang Shu},
  booktitle={Journal of Computational Physics},
Juan Cheng, and Chi-Wang Shu. Positivity-preserving Lagrangian scheme for multi-material compressible flow. 2014. Vol. 257. In Journal of Computational Physics. pp.143-168. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012044207619184161.
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