Positivity-Preserving High-Order Schemes for Conservation Laws on Arbitrarily Distributed Point Clouds with a Simple WENO Limiter
Year: 2018
Author: Jie Du, Chi-Wang Shu
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 1-2 : pp. 1–25
Abstract
This is an extension of our earlier work [9] in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. 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 [9] 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 [27], 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2018-IJNAM-10553
International Journal of Numerical Analysis and Modeling, Vol. 15 (2018), Iss. 1-2 : pp. 1–25
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: WENO limiter positivity-preserving arbitrarily distributed point cloud conservation laws high order.