A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations |
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Authors: | Zhiwei Cao Zhifeng Liu Xiaohong Wang Anfeng Shi Haishan Luo Benoît Noetinger |
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Institution: | 1. Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, Anhui, China;2. Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX, USA;3. IFP Energies nouvelles, Paris, France |
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Abstract: | In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy. |
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Keywords: | conservation laws method of characteristics nonlinear hyperbolic equation range‐discrete method shock wave simulation traffic flow |
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