Low-Dissipation 5th-Order Entropy Stable Schemes
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摘要: 双曲守恒律方程间断解的存在使其对数值求解格式的精度、分辨率等要求很高.Tadmor等构造的熵稳定(entropy stable, ES)格式，其数值解收敛到具有物理意义的唯一解，但耗散大，抹平严重，空间精度只有一阶.因此，将具有低数值耗散的TENO(targeted essentially non-oscillatory)重构引入到TeCNO框架中，构造出低耗散五阶TENO型熵稳定格式.证明了重构的熵变量在单元交界面处的跳跃满足保号性及所构造格式的熵稳定性.最后通过多种不同数值算例，检验五阶TENO型熵稳定格式的低数值耗散、高收敛阶、高分辨率及良好的数值鲁棒性.Abstract: The existence of intermittent solutions to hyperbolic conservation law equations requires high accuracy and resolution of the numerical solution schemes. The entropy stable schemes constructed by Tadmor et al. has numerical solutions that converge to physically meaningful unique solutions, but with severe dissipation large smearing effects and only 1st-order spatial accuracy. Therefore, the TENO (targeted essentially non-oscillatory) reconstruction with low numerical dissipation was introduced into the TeCNO framework, and a low-dissipation 5th-order TENO-type entropy stable scheme was constructed. It was proved that the jumps of the reconstructed entropy variables at the cell interfaces satisfy the sign-preserving property and the entropy stability of the constructed schemes. Finally, the low numerical dissipation, high convergence order, high resolution and good numerical robustness of the 5th-order TENO-type entropy stable scheme, were verified through various numerical examples.
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图 1 线性对流方程在间断初值下不同格式的结果
Figure 1. Results of linear advection equations in different schemes with interrupted initial values
图 2 无黏Burgers方程在间断初值下不同格式结果(左)与总熵变化(右)
Figure 2. Results (left) of different schemes and total entropy values (right) for inviscid Burgers' equation at interrupted initial values
图 3 溃坝问题不同格式结果(左)与总熵变化(右)
Figure 3. Results of different schemes (left) and total entropy values (right) under the dam failure problem
图 4 Sod激波管问题不同格式结果与总熵变化(右下)
Figure 4. Results of different schemes and total entropy values (lower right) under Sod's shock tube problem
图 5 Shu-Osher问题不同格式结果(左)与总熵变化(右)
Figure 5. Results of different schemes (left) and total entropy values (right) under the Shu-Osher test
图 6 间断初值下TENO(左)与熵相容(右)格式结果
Figure 6. Results of TENO (left) and EC (right) schemes with interrupted initial values
图 7 圆形溃坝问题TENO(左)与熵相容(右)格式结果
Figure 7. Results of TENO (left) and EC (right) schemes for the circular dam failure problem
图 8 聚焦激波管问题TENO(左)与熵相容(右)格式结果
Figure 8. Results of TENO (left) and EC (right) schemes for the focused shock tube problem
图 9 圆形溃坝问题(左)与聚焦激波管问题(右)立体图
Figure 9. The 3D views of the circular dam failure problem (left) and the focused shock tube problem (right)
图 10 Riemann问题Ⅰ(左)和Riemann问题Ⅱ(右)TENO型格式结果
Figure 10. TENO-type schemes's results for Riemann problem Ⅰ (left) and Riemann problem Ⅱ (right)
表 1 线性对流方程连续初值的误差及收敛阶
Table 1. Errors and orders of convergence of linear advection equations for continuous initial values
grid L1 error L1 order L2 error L2 order L∞ error L∞ order 32 1.937 718×10－5 - 2.148 915×10－5 - 3.027 211×10－5 - 64 6.076 331×10－7 4.995 0 6.746 401×10－7 4.993 3 9.529 882×10－7 4.989 4 128 1.900 491×10－8 4.998 8 2.110 875×10－8 4.998 2 2.985 020×10－8 4.996 7 256 5.943 347×10－10 4.999 0 6.601 338×10－10 4.998 9 9.335 437×10－10 4.998 9 512 1.866 531×10－11 4.992 8 2.072 894×10－11 4.993 0 2.930 567×10－11 4.993 5 
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表 2 无黏Burgers方程在连续初值下的误差及收敛阶
Table 2. Errors and orders of convergence of inviscid Burgers' equation for continuous initial values
grid L1 error L1 order L2 error L2 order L∞ error L∞ order 32 5.128 326×10－5 - 1.183 827×10－4 - 4.342 254×10－4 - 64 2.241 069×10－6 4.516 2 4.075 156×10－6 4.860 5 1.606 315×10－5 4.756 6 128 8.619 284×10－8 4.700 5 1.619 516×10－7 4.653 2 6.916 461×10－7 4.537 6 256 2.414 433×10－9 5.157 8 4.647 648×10－9 5.122 9 2.051 970×10－8 5.075 0 512 5.674 551×10－11 5.411 0 1.113 036×10－10 5.383 9 5.298 715×10－10 5.275 2 1 024 1.406 237×10－12 5.334 6 3.219 263×10－12 5.111 6 1.685 588×10－11 4.974 3
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[1] SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes[J]. Journal of Computational Physics, 1988, 77(2): 439-471. doi: 10.1016/0021-9991(88)90177-5 [2] LIU X D, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics, 1994, 115(1): 200-212. doi: 10.1006/jcph.1994.1187 [3] JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228. doi: 10.1006/jcph.1996.0130 [4] TADMOR E. The numerical viscosity of entropy stable schemes for systems of conservation laws, Ⅰ[J]. Mathematics of Computation, 1987, 49(179): 91-103. doi: 10.1090/S0025-5718-1987-0890255-3 [5] ISMAIL F, ROE P L. Affordable, entropy-consistent Euler flux functions Ⅱ: entropy production at shocks[J]. Journal of Computational Physics, 2009, 228(15): 5410-5436. doi: 10.1016/j.jcp.2009.04.021 [6] FJORDHOLM U S, MISHRA S, TADMOR E. Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws[J]. SIAM Journal on Numerical Analysis, 2012, 50(2): 544-573. doi: 10.1137/110836961 [7] FJORDHOLM U S, RAY D. A sign preserving WENO reconstruction method[J]. Journal of Scientific Computing, 2016, 68: 42-63. doi: 10.1007/s10915-015-0128-y [8] BISWAS B, DUBEY R K. Low dissipative entropy stable schemes using third order WENO and TVD reconstructions[J]. Advances in Computational Mathematics, 2018, 44: 1153-1181. doi: 10.1007/s10444-017-9576-2 [9] 郑素佩, 赵青宇, 封建湖. 基于WENO重构保号的四阶熵稳定格式[J]. 浙江大学学报(理学版), 2022, 49(3): 329-335.ZHENG Supei, ZHAO Qingyu, FENG Jianhu. The fourth order entropy stable scheme based on sign-preserving WENO reconstruction[J]. Journal of Zhejiang University (Science Edition), 2022, 49(3): 329-335. (in Chinese) [10] 郑素佩, 徐霞, 封建湖, 等. 高阶保号熵稳定格式[J]. 数学物理学报, 2021, 41(5): 1296-1310.ZHENG Supei, XU Xia, FENG Jianhu, et al. High order sign preserving entropy stable schemes[J]. Acta Mathematica Scientia, 2021, 41(5): 1296-1310. (in Chinese) [11] 郑素佩, 建芒芒, 封建湖, 等. 保号WENO-AO型中心迎风格式[J]. 计算物理, 2022, 39(6): 677-686.ZHENG Supei, JIAN Mangmang, FENG Jianhu, et al. Sign preserving WENO-AO-type central upwind schemes[J]. Chinese Journal of Computational Physics, 2022, 39(6): 677-686. (in Chinese) [12] 张成治, 郑素佩, 陈雪, 等. 求解理想磁流体方程的四阶WENO型熵稳定格式[J]. 应用数学和力学, 2023, 44(11): 1398-1412. doi: 10.21656/1000-0887.440178ZHANG Chengzhi, ZHENG Supei, CHEN Xue, et al. A 4th-order WENO-type entropy stable scheme for ideal magnetohydrodynamic equations[J]. Applied Mathematics and Mechanics, 2023, 44(11): 1398-1412. (in Chinese) doi: 10.21656/1000-0887.440178 [13] FU L, HU X Y, ADAMS N A. A family of high-order targeted ENO schemes for compressible-fluid simulations[J]. Journal of Computational Physics, 2016, 305: 333-359. doi: 10.1016/j.jcp.2015.10.037 [14] FU L. Review of the high-order TENO schemes for compressible gas dynamics and turbulence[J]. Archives of Computational Methods in Engineering, 2023, 30(4): 2493-2526. doi: 10.1007/s11831-022-09877-7 [15] GOTTLIEB S, SHU C W, TADMOR E. Strong stability-preserving high-order time discretization methods[J]. SIAM Review, 2001, 43(1): 89-112. doi: 10.1137/S003614450036757X [16] GEROLYMOS G A, SÉNÉCHAL D, VALLET I. Very-high-order WENO schemes[J]. Journal of Computational Physics, 2009, 228(23): 8481-8524. doi: 10.1016/j.jcp.2009.07.039 [17] LAX P D. Weak solutions of nonlinear hyperbolic equations and their numerical computation[J]. Communications on Pure and Applied Mathematics, 1954, 7(1): 159-193. doi: 10.1002/cpa.3160070112 [18] LEFLOCH P G, MERCIER J M, ROHDE C. Fully discrete, entropy conservative schemes of arbitrary order[J]. SIAM Journal on Numerical Analysis, 2002, 40(5): 1968-1992. doi: 10.1137/S003614290240069X -
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