Positivity-preserving high-order schemes for conservation laws on arbitrarily distributed point clouds with a simple WENO limiter

Jie Du Tsinghua University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25019

International Journal of Numerical Analysis and Modeling, 15, 1-25, 2018
This is an extension of our earlier work \cite{DS} in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. %The initial condition is given on such a point cloud, %and the algorithm solves for point values of the solution at %later time also on this point cloud. An algorithm of building a suitable polygonal mesh based on the random points was given and the traditional discontinuous Galerkin (DG) method was adopted on the constructed polygonal mesh. Numerical results in \cite{DS} show that the current scheme will generate spurious numerical oscillations when dealing with solutions containing strong shocks. In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for DG schemes on two-dimensional unstructured triangular meshes \cite{ZZSQ}, to our high order method on polygonal meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the original method in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt is particularly simple to implement and will not harm the conservativeness and compactness of the original method. Moreover, we also extend the maximum-principle-satisfying limiter for the scalar case and the positivity-preserving limiter for the Euler system to our method. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of these limiters.
WENO limiter, positivity-preserving, arbitrarily distributed point cloud, conservation laws, high order
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@inproceedings{jie2018positivity-preserving,
  title={Positivity-preserving high-order schemes for conservation laws on arbitrarily distributed point clouds with a simple WENO limiter},
  author={Jie Du, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416103611325759054},
  booktitle={International Journal of Numerical Analysis and Modeling},
  volume={15},
  pages={1-25},
  year={2018},
}
Jie Du, and Chi-Wang Shu. Positivity-preserving high-order schemes for conservation laws on arbitrarily distributed point clouds with a simple WENO limiter. 2018. Vol. 15. In International Journal of Numerical Analysis and Modeling. pp.1-25. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416103611325759054.
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