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Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods

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Abstract

A cut cell based sharp-interface Runge–Kutta discontinuous Galerkin method, with quadtree-like adaptive mesh refinement, is developed for simulating compressible two-medium flows with clear interfaces. In this approach, the free interface is represented by curved cut faces and evolved by solving the level-set equation with high order upstream central scheme. Thus every mixed cell is divided into two cut cells by a cut face. The Runge–Kutta discontinuous Galerkin method is applied to calculate each single-medium flow governed by the Euler equations. A two-medium exact Riemann solver is applied on the cut faces and the Lax–Friedrichs flux is applied on the regular faces. Refining and coarsening of meshes occur according to criteria on distance from the material interface and on magnitudes of pressure/density gradient, and the solutions and fluxes between upper-level and lower-level meshes are synchronized by \(L^2\) projections to keep conservation and high order accuracy. This proposed method inherits the advantages of the discontinuous Galerkin method (compact and high order) and cut cell method (sharp interface and curved cut face), thus it is fully conservative, consistent, and is very accurate on both interface and flow field calculations. Numerical tests with a variety of parameters illustrate the accuracy and robustness of the proposed method.

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Acknowledgements

The authors acknowledge the supports from National Science Foundation of China (NSFC, Grant Nos. 91230203, 11202020 and U1530401) and President Foundation of Chinese Academy of Engineering Physics (Grant No. 201501043). Dr. Maojun Li is partially supported by the NSFC (Grant Nos. 11501062, 91630205).

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Xiao-Long Deng & Maojun Li

  2. Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA, 22904, USA

    Xiao-Long Deng

  3. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Maojun Li

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Correspondence to Xiao-Long Deng.

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Deng, XL., Li, M. Simulating Compressible Two-Medium Flows with Sharp-Interface Adaptive Runge–Kutta Discontinuous Galerkin Methods. J Sci Comput 74, 1347–1368 (2018). https://doi.org/10.1007/s10915-017-0511-y

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