EI、Scopus 收录
中文核心期刊

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用

钟巍 贾雷明 王澍霏 田宙

钟巍, 贾雷明, 王澍霏, 田宙. 一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用. 力学学报, 2022, 54(11): 1-22 doi: 10.6052/0459-1879-22-247
引用本文: 钟巍, 贾雷明, 王澍霏, 田宙. 一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用. 力学学报, 2022, 54(11): 1-22 doi: 10.6052/0459-1879-22-247
Zhong Wei, Jia Leiming, Wang Shufei, Tian Zhou. A high-efficiency and high-resolution mapped weno scheme and its applications in the numerical simulation of problems with complex flows. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 1-22 doi: 10.6052/0459-1879-22-247
Citation: Zhong Wei, Jia Leiming, Wang Shufei, Tian Zhou. A high-efficiency and high-resolution mapped weno scheme and its applications in the numerical simulation of problems with complex flows. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 1-22 doi: 10.6052/0459-1879-22-247

一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用

doi: 10.6052/0459-1879-22-247
详细信息
    作者简介:

    钟巍, 副研究员/副主任, 主要研究方向: 计算流体力学. E-mail: zhongwei2016@pku.edu.cn

  • 中图分类号: O241.82

A HIGH-EFFICIENCY AND HIGH-RESOLUTION MAPPED WENO SCHEME AND ITS APPLICATIONS IN THE NUMERICAL SIMULATION OF PROBLEMS WITH COMPLEX FLOWS

  • 摘要: 由于映射操作会带来额外的计算时间消耗, 传统加映射的WENO格式存在计算效率低的缺陷. 为了提高传统加映射WENO格式的计算效率, 通过利用标准符号函数的一种近似逼近函数构造出一族近似常值映射函数, 本文提出了一种新的加映射WENO格式, 称为WENO-ACM. 新映射函数满足文献中已有WENO-PM6格式映射函数的全部设计要求, 其中WENO-PM6是一种为了克服经典WENO-M格式潜在的精度丢失缺陷而提出的格式. 新格式保留了WENO-PM6在低耗散和高分辨率方面的优势, 同时, 显著的减少了每个时间步映射过程中的数学运算操作数, 进而在计算效率方面获得了明显的提升. 理论分析表明, 新格式在即使包含临界点的光滑区域也能够获得最佳收敛精度. 对近似色散关系的研究表明, 新格式的频谱特性也得到了显著的提升. 对大量标准测试算例进行了模拟计算, 包括精度测试、激波管问题、激波-熵波相互作用、爆炸波相互碰撞、二维黎曼问题、双马赫反射、前台阶流动、瑞利-泰勒不稳定性和开尔文-亥姆霍兹不稳定性问题等. 与广泛认可的WENO-JS、WENO-M和WENO-PM6格式综合比较发现, 新提出的WENO-ACM格式在高效率、低数值耗散和间断捕捉等方面都有显著的改进. 最重要的是, 与WENO-M和WENO-PM6格式相比, WENO-ACM将相对于WENO-JS格式的额外计算时间消耗分别减少了80%和90%以上.

     

  • 图  1  WENO-ACM格式的映射函数曲线, $ {C_1} = 0.6 $

    Figure  1.  Mapping functions of WENO-ACM, $ {C_1} = 0.6 $

    图  2  不同WENO格式的频谱特性曲线比较

    Figure  2.  Comparison of the spectral properties of different WENO schemes

    图  3  不同WENO格式对初始条件1的计算误差与CPU计算时间关系曲线

    Figure  3.  Comparison of various WENO schemes for initial condition 1 in CPU time and numerical computing errors

    图  4  不同WENO格式对初始条件2的计算误差与CPU计算时间关系曲线

    Figure  4.  Comparison of various WENO schemes for initial condition 2 in CPU time and numerical computing errors

    图  5  不同WENO格式对包含高阶临界点光滑问题的长时间模拟结果, $ N = 300,t = 1200 $

    Figure  5.  Results of various WENO schemes for the smooth problem with high-order critical points at long output time, $ N = 300,t = 1200 $

    图  6  不同WENO格式对高阶临界点光滑问题的计算误差与CPU计算时间关系曲线

    Figure  6.  Comparison of various WENO schemes for the smooth problem with high-order critical points in CPU time and numerical computing errors

    图  7  不同WENO格式对Sod和Lax激波管问题的模拟结果

    Figure  7.  Results of various WENO schemes for the Sod and Lax shock-tube problems

    图  8  不同WENO格式对Woodward-Colella爆炸波碰撞问题的模拟结果

    Figure  8.  Results of various WENO schemes for the Woodward-Colella blast wave interacting problem

    图  9  不同WENO格式对Shu-Osher激波-熵波相互作用问题的模拟结果

    Figure  9.  Results of various WENO schemes for the Shu-Osher shock-entropy interaction problem

    图  10  不同WENO格式对Titarev-Toro激波-熵波相互作用问题的模拟结果

    Figure  10.  Results of various WENO schemes for the Titarev-Toro shock-entropy interaction problem

    图  11  不同WENO格式对二维黎曼问题的模拟结果

    Figure  11.  Results of various WENO schemes for the 2 D Riemann problem

    图  12  不同WENO格式对双马赫反射问题模拟结果

    Figure  12.  Results of various WENO schemes for the double Mach reflection problem

    图  13  不同WENO格式对前台阶流问题的模拟结果, 900 × 300个网格

    Figure  13.  Results of various WENO schemes for the forward facing step problem, 900 × 300 cells.

    图  14  不同WENO格式对前台阶流问题的模拟结果, 1800 × 600个网格

    Figure  14.  Results of various WENO schemes for the forward facing step problem, 1800 × 600 cells

    图  15  不同WENO格式对瑞利-泰勒不稳定性问题的模拟结果

    Figure  15.  Results of various WENO schemes for the Rayleigh-Taylor instability

    图  16  不同WENO格式对开尔文-亥姆霍兹不稳定性问题的模拟结果

    Figure  16.  Results of various WENO schemes for the Kelvin-Helmholtz instability

    图  17  竖直平面激波与热层作用示意图

    Figure  17.  Illustration of the interaction of a vertical planar shock with a thermal layer

    图  18  Autodyn和WENO-ACM计算所得流场密度分布

    Figure  18.  Density distribution computed by Autodyn and WENO-ACM

    图  19  Y = 4截面上的密度变化曲线比较

    Figure  19.  The cross-sectional slices of density plot along the plane Y = 4

    表  1  不同WENO格式的$ {L^2} $$ {L^\infty } $误差及其收敛精度, 初始条件1

    Table  1.   $ {L^2} $ and $ {L^\infty } $ errors and the convergence properties of various schemes for initial condition 1

    NWENO-JS
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    202.43 × 10−32.58 × 10−3
    407.64 × 10−54.999.05 × 10−54.83
    802.34 × 10−65.032.91 × 10−64.96
    1607.19 × 10−85.028.86 × 10−85.04
    3202.23 × 10−95.012.72 × 10−95.02
    NWENO-M
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    204.06 × 10−43.95 × 10−4
    401.25 × 10−55.021.25 × 10−54.98
    803.92 × 10−75.003.92 × 10−74.99
    1601.23 × 10−85.001.23 × 10−85.00
    3203.84 × 10−105.003.84 × 10−105.00
    NWENO-PM6
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    203.95 × 10−43.95 × 10−4
    401.25 × 10−54.981.25 × 10−54.98
    803.92 × 10−74.993.92 × 10−74.99
    1601.23 × 10−85.001.23 × 10−85.00
    3203.84 × 10−105.003.84 × 10−105.00
    NWENO-ACM
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    203.95 × 10−43.94 × 10−4
    401.25 × 10−54.981.25 × 10−54.98
    803.92 × 10−74.993.92 × 10−74.99
    1601.23 × 10−85.001.23 × 10−85.00
    3203.84 × 10−105.003.84 × 10−105.00
    下载: 导出CSV

    表  2  不同WENO格式的$ {L^2} $$ {L^\infty } $误差及其收敛精度, 初始条件2

    Table  2.   $ {L^2} $ and $ {L^\infty } $ errors and the convergence properties of various schemes for initial condition 2

    NWENO-JS
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    208.72 × 10−31.43 × 10−2
    406.76 × 10−43.691.10 × 10−33.71
    803.64 × 10−54.229.02 × 10−53.60
    1602.29 × 10−63.998.24 × 10−63.45
    3201.68 × 10−73.778.32 × 10−73.31
    NWENO-M
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    203.36 × 10−35.44 × 10−3
    401.39 × 10−44.592.19 × 10−44.64
    804.53 × 10−64.946.81 × 10−65.00
    1601.42 × 10−74.992.15 × 10−74.99
    3204.46 × 10−94.996.71 × 10−94.99
    NWENO-PM6
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    204.29 × 10−35.91 × 10−3
    401.44 × 10−44.902.10 × 10−44.82
    804.54 × 10−64.986.83 × 10−64.94
    1601.42 × 10−74.992.15 × 10−74.99
    3204.46 × 10−95.006.71 × 10−95.00
    NWENO-ACM
    $ {L^2} $−error$ {L^2} $−order$ {L^\infty } $−error$ {L^\infty } $−order
    204.03 × 10−35.89 × 10−3
    401.42 × 10−44.822.10 × 10−44.81
    804.54 × 10−64.976.83 × 10−64.94
    1601.42 × 10−74.992.14 × 10−74.99
    3204.46 × 10−95.006.71 × 10−95.00
    下载: 导出CSV

    表  3  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 二维黎曼问题

    Table  3.   CPU time and the extra computational cost per Runge-Kutta step for the 2 D Riemann problem

    200 × 200 grids (unit: second)
    Test 1Test 2Test 3Average
    JS0.22(-)0.21(-)0.22(-)0.22(-)
    M0.28(29%)0.27(24%)0.28(28%)0.28(27%)
    PM60.34(57%)0.33(53%)0.34(56%)0.34(55%)
    ACM0.23(4%)0.22(5%)0.23(5%)0.23(5%)
    ηM86%80%80%82%
    ηPM693%91%90%91%
    400 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    JS0.83(-)0.82(-)0.83(-)0.83(-)
    M1.04(26%)1.03(26%)1.02(24%)1.03(25%)
    PM61.27(53%)1.28(56%)1.30(57%)1.28(55%)
    ACM0.86(4%)0.84(3%)0.87(4%)0.86(4%)
    ηM85%89%81%85%
    ηPM693%95%92%93%
    下载: 导出CSV

    表  4  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 双马赫反射问题

    Table  4.   CPU time and the extra computational cost per Runge-Kutta step for the double Mach reflection problem

    1000 × 250 grids (unit: second)
    Test 1Test 2Test 3Average
    JS1.13(-)1.16(-)1.16(-)1.15(-)
    M1.46(29%)1.48(27%)1.47(27%)1.47(28%)
    PM61.79(58%)1.76(51%)1.77(53%)1.77(54%)
    ACM1.18(4%)1.19(3%)1.19(3%)1.19(3%)
    ηM86%90%90%88%
    ηPM693%95%95%94%
    2000 × 500 grids (unit: second)
    Test 1Test 2Test 3Average
    JS4.30(-)4.28(-)4.31(-)4.30(-)
    M5.61(31%)5.62(31%)5.65(31%)5.63(31%)
    PM66.75(57%)6.74(58%)6.77(57%)6.75(57%)
    ACM4.46(4%)4.44(4%)4.50(5%)4.47(4%)
    ηM88%88%85%87%
    ηPM693%94%92%93%
    下载: 导出CSV

    表  5  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 前台阶流问题

    Table  5.   CPU time and the extra computational cost per Runge-Kutta step for the forward facing step problem

    900 × 300 grids (unit: second)
    Test 1Test 2Test 3Average
    JS1.87(-)1.88(-)1.87(-)1.87(-)
    M2.50(33%)2.50(33%)2.47(32%)2.49(33%)
    PM63.14(68%)3.12(66%)3.15(68%)3.14(67%)
    ACM1.95(4%)1.93(2%)1.93(3%)1.93(3%)
    ηM88%93%90%90%
    ηPM694%96%95%95%
    1200 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    JS3.22(-)3.24(-)3.23(-)3.23(-)
    M4.31(34%)4.36(35%)4.29(33%)4.32(34%)
    PM65.37(67%)5.50(70%)5.42(68%)5.43(68%)
    ACM3.36(4%)3.39(5%)3.38(5%)3.38(5%)
    ηM87%86%86%86%
    ηPM694%93%93%93%
    下载: 导出CSV

    表  6  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 瑞利-泰勒不稳定性问题

    Table  6.   CPU time and the extra computational cost per Runge-Kutta step for the Rayleigh-Taylor instability

    120 × 480 grids (unit: second)
    Test 1Test 2Test 3Average
    JS0.38(-)0.39(-)0.39(-)0.39(-)
    M0.48(27%)0.49(26%)0.49(26%)0.49(26%)
    PM60.61(60%)0.60(54%)0.60(55%)0.60(56%)
    ACM0.39(3%)0.40(3%)0.40(3%)0.40(3%)
    ηM90%87%88%88%
    ηPM696%94%94%95%
    240 × 960 grids (unit: second)
    Test 1Test 2Test 3Average
    JS1.53(-)1.55(-)1.54(-)1.54(-)
    M1.97(28%)1.95(26%)1.96(27%)1.96(27%)
    PM62.36(54%)2.39(54%)2.38(54%)2.38(54%)
    ACM1.59(4%)1.58(2%)1.58(3%)1.59(3%)
    ηM86%92%90%90%
    ηPM693%96%95%95%
    下载: 导出CSV

    表  7  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 开尔文-亥姆霍兹不稳定性问题

    Table  7.   CPU time and the extra computational cost per Runge-Kutta step for the Kelvin-Helmholtz instability

    256 × 256 grids (unit: second)
    Test 1Test 2Test 3Average
    JS0.43(-)0.44(-)0.43(-)0.43(-)
    M0.54(27%)0.55(25%)0.54(24%)0.54(25%)
    PM60.67(56%)0.70(60%)0.68(58%)0.68(58%)
    ACM0.45(4%)0.45(4%)0.46(5%)0.45(4%)
    ηM84%85%80%83%
    ηPM692%94%92%93%
    512 × 512 grids (unit: second)
    Test 1Test 2Test 3Average
    JS1.67(-)1.67(-)1.68(-)1.68(-)
    M2.12(27%)2.14(28%)2.12(26%)2.13(27%)
    PM62.67(60%)2.64(58%)2.64(57%)2.65(58%)
    ACM1.74(4%)1.74(4%)1.74(3%)1.74(4%)
    ηM84%86%87%86%
    ηPM693%93%94%93%
    下载: 导出CSV

    表  8  每个Runge-Kutta步的CPU计算时间分布情况, 二维黎曼问题

    Table  8.   CPU time allocation per Runge-Kutta step for the 2 D Riemann problem

    JS, 400 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.203(24%)0.202(25%)0.201(24%)0.202(24%)
    Trec0.563(68%)0.558(68%)0.572(68%)0.564(68%)
    Tmap0.000(0%)0.000 (0%)0.000 (0%)0.000 (0%)
    Toth0.063(8%)0.063(8%)0.064(8%)0.063(8%)
    Tall0.829(100%)0.823(100%)0.837(100%)0.830(100%)
    M, 400 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.188(0.18%)0.193(19%)0.203(20%)0.195(19%)
    Trec0.529(51%)0.517(50%)0.482(47%)0.509(50%)
    Tmap0.252(24%)0.264(25%)0.268(26%)0.261(25%)
    Toth0.062(6%)0.063(6%)0.063(6%)0.063(6%)
    Tall1.031(100%)1.037(100%)1.016(100%)1.028(100%)
    PM6, 400 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.188(15%)0.202(16%)0.203(16%)0.198(15%)
    Trec0.516(41%)0.453(35%)0.546(42%)0.505(39%)
    Tmap0.499(39%)0.563(44%)0.485(37%)0.516(40%)
    Toth0.062(5%)0.063(5%)0.062(5%)0.062(5%)
    Tall1.265(100%)1.281(100%)1.296(100%)1.281(100%)
    ACM, 400 × 400 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.204(24%)0.188(22%)0.204(23%)0.199(23%)
    Trec0.531(63%)0.546(65%)0.577(66%)0.551(65%)
    Tmap0.046(5%)0.047(6%)0.032(4%)0.042(5%)
    Toth0.062(7%)0.063(7%)0.062(7%)0.062(7%)
    Tall0.843(100%)0.844(100%)0.875(100%)0.854(100%)
    下载: 导出CSV

    表  9  每个Runge-Kutta步的CPU计算时间分布情况, 双马赫反射问题

    Table  9.   CPU time allocation per Runge-Kutta step for the double Mach reflection problem

    JS, 1000 × 250 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.250(22%)0.306(25%)0.266(23%)0.274(23%)
    Trec0.828(73%)0.861(70%)0.825(71%)0.838(71%)
    Tmap0.000(0%)0.000(0%)0.000(0%)0.000(0%)
    Toth0.063(6%)0.063(5%)0.063(5%)0.062(5%)
    Tall1.141(100%)1.230(100%)1.154(100%)1.175(100%)
    M, 1000 × 250 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.234(16%)0.234(16%)0.249(17%)0.239(16%)
    Trec0.755(52%)0.765(53%)0.797(54%)0.772(53%)
    Tmap0.401(28%)0.391(27%)0.375(25%)0.389(27%)
    Toth0.063(4%)0.063(4%)0.063(4%)0.063(4%)
    Tall1.453(100%)1.453(100%)1.484(100%)1.463(100%)
    PM6, 1000 × 250 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.234(13%)0.219(12%)0.235(13%)0.229(13%)
    Trec0.781(44%)0.719(41%)0.707(40%)0.736(42%)
    Tmap0.703(39%)0.765(43%)0.761(43%)0.743(42%)
    Toth0.063(4%)0.063(4%)0.062(4%)0.063(4%)
    Tall1.781(100%)1.766(100%)1.765(100%)1.771(100%)
    ACM, 1000 × 250 grids (unit: second)
    Test 1Test 2Test 3Average
    Tch0.250(21%)0.266(22%)0.251(21%)0.256(22%)
    Trec0.798(67%)0.843(71%)0.828(71%)0.823(70%)
    Tmap0.076(6%)0.016(1%)0.031(3%)0.041(3%)
    Toth0.063(5%)0.062(5%)0.062(5%)0.062(5%)
    Tall1.187(100%)1.187(100%)1.172(100%)1.182(100%)
    下载: 导出CSV

    表  10  每个Runge-Kutta步的CPU计算时间及相对WENO-JS的额外计算时间, 竖直平面激波与热层作用问题

    Table  10.   CPU time and the extra computational cost per Runge-Kutta step for the interaction of a vertical planar shock with a thermal layer

    500 × 500 grids (unit: second)
    Test 1Test 2Test 3Average
    JS1.25(-)1.24(-)1.27(-)1.25(-)
    M1.68(34%)1.69(36%)1.67(31%)1.68(34%)
    PM62.09(67%)2.20(77%)2.11(66%)2.13(70%)
    ACM1.31(4%)1.27(3%)1.30(3%)1.29(3%)
    ηM88%93%91%91%
    ηPM694%97%96%95%
    下载: 导出CSV
  • [1] Jiang GS, Shu CW. Efficient implementation of weighted ENO schemes. Journal of Computational Physics, 1996, 126: 202-228 doi: 10.1006/jcph.1996.0130
    [2] Huang ZY, Lin G, Ardekani AM. A mixed upwind/central WENO scheme for incompressible two-phase flows. Journal of Computational Physics, 2019, 387: 455-480 doi: 10.1016/j.jcp.2019.02.043
    [3] Liu HP, Gao ZX, Jiang CW, et al. Numerical study of combustion effects on the development of supersonic turbulent mixing layer flows with WENO schemes. Computers and Fluids, 2019, 189: 82-93 doi: 10.1016/j.compfluid.2019.05.019
    [4] Kumar S, Singh P. High order WENO finite volume approximation for population density neuron model. Applied Mathematics and Computation, 2019, 356: 173-189 doi: 10.1016/j.amc.2019.03.020
    [5] Lefevre V, Garnica A, Lopez-Pamies O. A WENO finite-difference scheme for a new class of Hamilton-Jacobi equations in nonlinear solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2019, 349: 173-189
    [6] Wang D, Byambaakhuu T. High-order lax-friedrichs WENO fast sweeping methods for the S N neutron transport equation. Nuclear Science and Engineering, 2019, 193(9): 982-990 doi: 10.1080/00295639.2019.1582316
    [7] Fu L, Tang Q. High-order low-dissipation targeted ENO schemes for ideal magnetohydrodynamics. Journal of Scientific Computing, 2019, 80: 692-716 doi: 10.1007/s10915-019-00941-2
    [8] Nonomura T, Fujii K. Characteristic finite-difference WENO scheme for multicomponent compressible fluid analysis: Overestimated quasi-conservative formulation maintaining equilibriums of velocity, pressure, and temperature. Journal of Computational Physics, 2017, 340: 358-388 doi: 10.1016/j.jcp.2017.02.054
    [9] 童福林, 李新亮, 唐志共. 激波与转捩边界层干扰非定常特性数值分析. 力学学报, 2017, 49(1): 93-104 doi: 10.6052/0459-1879-16-224

    Tong Fulin, Li Xinliang, Tang Zhigong. Numerical analysis of unsteady motion in shock wave/transitional boundary layer interaction. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 93-104 (in Chinese) doi: 10.6052/0459-1879-16-224
    [10] 童福林, 李欣, 于长平等. 高超声速激波湍流边界层干扰直接数值模拟研究. 力学学报, 2018, 50(2): 197-208 doi: 10.6052/0459-1879-17-239

    Tong Fulin, Li Xin, Yu Changping, Li Xinliang. Direct numerical simulation of hypersonic shock wave and turbulent boundary layer interactions. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 197-208 (in Chinese) doi: 10.6052/0459-1879-17-239
    [11] Duan JM, Tang HZ. High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics. Journal of Computational Physics, 2021, 431: 110136 doi: 10.1016/j.jcp.2021.110136
    [12] Duan JM, Tang HZ. An efficient ADER discontinuous Galerkin scheme for directly solving Hamilton-Jacobi equation. Journal of Computational Mathematics, 2020, 38(1): 58-83 doi: 10.4208/jcm.1902-m2018-0189
    [13] Balsara DS, Meyer C, Dumbser M, et al. Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes-Speed comparisons with Runge-Kutta methods. Journal of Computational Physics, 2013, 235: 934-969 doi: 10.1016/j.jcp.2012.04.051
    [14] Balsara DS, Dumbser M, Abgrall R. Multidimensional HLLC Riemann solver for unstructured meshes-with application to Euler and MHD flows. Journal of Computational Physics, 2014, 261: 172-208 doi: 10.1016/j.jcp.2013.12.029
    [15] 贾雷明, 田宙. 竖直平面激波与水平热层作用后流场的理论计算方法研究. 爆炸与冲击, 2019, 39(12): 122202-1-11 doi: 10.11883/bzycj-2018-0510

    Jia Leiming, Tian Zhou. On the theoretical calculation method for interaction between the vertical plane shock wave and the horizontal thermal layer. Explosion and Shock Waves, 2019, 39(12): 122202-1-11 (in Chinese) doi: 10.11883/bzycj-2018-0510
    [16] 张树海. 加权紧致格式与 WENO 格式的比较研究. 力学学报, 2016, 48(2): 336-347 (Zhang Shuhai. The comparison of weighted compact schemes and WENO scheme. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2): 336-347 (in Chinese)) doi: 10.6052/0459-1879-19-249

    Luo Xin, Wu Songping. An improved fifth-order WENO-Z + scheme. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1927-1939 (in Chinese) doi: 10.6052/0459-1879-19-249
    [17] 张树海. 加权紧致格式与 WENO 格式的比较研究. 力学学报, 2016, 48(2): 336-347

    Zhang Shuhai. The comparison of weighted compact schemes and WENO scheme. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2): 336-347 (in Chinese))
    [18] 韦志龙, 蒋勤. 基于 WENO-THINC/WLIC 模型的水气二相流数值模拟. 力学学报, 2021, 53(4): 973-985 doi: 10.6052/0459-1879-20-430

    Wei Zhilong, Jiang Qin. Numerical study on water-air two-phase flow based on WENO-THINC/WLIC model. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(4): 973-985 (in Chinese) doi: 10.6052/0459-1879-20-430
    [19] 崔竹轩, 丁举春, 司廷. 反射激波作用下三维凹气柱界面演化的数值研究. 力学学报, 2021, 53(5): 1246-1256 (Cui Zhuxuan, Ding Juchun, Si Ting. Numerical study on the evolution of three-dimensional concave cylindrical interface accelerated by reflected shock . Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(5): 1246-1256 (in Chinese)) doi: 10.6052/0459-1879-14-089

    Liu Yu, Liu Jun, Tang Lingyan, Cui Xiaoqiang. A novel uncoupled algorithm for solving chemical nonequilibrium flows. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(1): 82-94. (in Chinese) doi: 10.6052/0459-1879-14-089
    [20] 崔竹轩, 丁举春, 司廷. 反射激波作用下三维凹气柱界面演化的数值研究. 力学学报, 2021, 53(5): 1246-1256

    Cui Zhuxuan, Ding Juchun, Si Ting. Numerical study on the evolution of three-dimensional concave cylindrical interface accelerated by reflected shock . Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(5): 1246-1256 (in Chinese))
    [21] Zhao WG, Zheng HW, Liu FJ, et al. An efficient unstructured WENO method for supersonic reactive flows. Acta Mechanica Sinica, 2018, 34(4): 623-631 doi: 10.1007/s10409-018-0756-1
    [22] Huang WF, Ren YX, Jiang X. A simple algorithm to improve the performance of the WENO scheme on non-uniform grids. Acta Mechanica Sinica, 2018, 34(1): 37-47 doi: 10.1007/s10409-017-0715-2
    [23] 余俊, 刘国振, 江俊, 等. 水下爆炸气泡射流载荷计算的新方法研究. 计算力学学报, 2021, 38(1): 120-125 doi: 10.7511/jslx20200524001

    Yu Jun, Liu Guozhen, Jiang Jun, et al. Research on a new method of bubble jet load in underwater explosion. Chinese Journal of Computational Mechanics, 2021, 38(1): 120-125 (in Chinese) doi: 10.7511/jslx20200524001
    [24] 徐维铮, 孔祥韶, 郑成, 等. 三阶WENO格式精度分析与改进. 计算力学学报, 2018, 35(2): 51-55 (Xu Weizheng, Kong Xiangzhao, Zheng Cheng, et al. Precision analysis and improvement of third-order WENO scheme. Chinese Journal of Computational Mechanics, 2018, 35(2): 51-55 (in Chinese)) doi: 10.7511/jslx20181113003

    Xu Li, Jiang Mingyang, Cai Jingjing, et al. On improved high accuracy scheme for compressible inviscid flow computation. Chinese Journal of Computational Mechanics, 2019, 36(6): 784-791 (in Chinese) doi: 10.7511/jslx20181113003
    [25] 徐维铮, 孔祥韶, 郑成, 等. 三阶WENO格式精度分析与改进. 计算力学学报, 2018, 35(2): 51-55

    Xu Weizheng, Kong Xiangzhao, Zheng Cheng, et al. Precision analysis and improvement of third-order WENO scheme. Chinese Journal of Computational Mechanics, 2018, 35(2): 51-55 (in Chinese))
    [26] Liu XD, Osher S, Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics, 1994, 115: 200-212 doi: 10.1006/jcph.1994.1187
    [27] Henrick AK, Aslam TD, Powers JM. Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. Journal of Computational Physics, 2005, 207: 542-567
    [28] Feng H, Hu FX, Wang R. A new mapped weighted essentially non-oscillatory scheme. Journal of Scientific Computing, 2012, 51: 449-473 doi: 10.1007/s10915-011-9518-y
    [29] Feng H, Huang C, Wang R. An improved mapped weighted essentially non-oscillatory scheme. Applied Mathematics and Computation, 2014, 232: 453-468 doi: 10.1016/j.amc.2014.01.061
    [30] Li Q, Liu PX, Zhang HX. Piecewise polynomial mapping method and corresponding WENO scheme with improved resolution. Communications in Computational Physics, 2015, 18(5): 1417-1444 doi: 10.4208/cicp.150215.250515a
    [31] Li R, Zhong W. A modified adaptive improved mapped WENO method. Communications in Computational Physics, 2021, 30(5): 1545-1588 doi: 10.4208/cicp.OA-2021-0057
    [32] Wang R, Feng H, Huang C. A new mapped weighted essentially non-oscillatory method using rational function. Journal of Scientific Computing, 2016, 67: 540-580 doi: 10.1007/s10915-015-0095-3
    [33] Borges R, Carmona M, Costa B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. Journal of Computational Physics, 2008, 227(6): 3191-3211 doi: 10.1016/j.jcp.2007.11.038
    [34] Zhang R, Zhang M P, Shu C W. On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Communications in Computational Physics, 2011, 9(3): 807-827 doi: 10.4208/cicp.291109.080410s
    [35] Gottlieb S, Shu CW. Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 1998, 67: 73-85 doi: 10.1090/S0025-5718-98-00913-2
    [36] Gottlieb S, Shu CW, Tadmor E. Strong stability-preserving high-order time discretization methods. SIAM Review, 2001, 43: 89-112 doi: 10.1137/S003614450036757X
    [37] Shu CW, Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. Journal of Computational Physics, 1988, 77: 439-471 doi: 10.1016/0021-9991(88)90177-5
    [38] Pirozzoli S. On the spectral properties of shock-capturing schemes. Journal of Computational Physics, 2006, 219(2): 489-497 doi: 10.1016/j.jcp.2006.07.009
    [39] 涂国华, 邓小刚, 毛枚良. 5阶非线性WCNS和WENO差分格式频谱特性比较. 空气动力学学报, 2012, 30(6): 709-712

    Tu Guo-hua, Deng Xiaogang, Mao Meiliang. Spectral property comparison of fifth-order nonlinear WCNS and WENO difference schemes. Acta Aerodynamica Sinica, 2012, 30(6): 709-712 (in Chinese)
    [40] 张瑞. 有限体积 WENO 格式及其应用. [博士论文]. 合肥: 中国科学技术大学, 2010: 29-50

    Zhang Rui. Finite Volume WENO Schemes and Applications. [PhD Thesis]. Hefei: University of Science and Technology of China, 2010: 29-50 (in Chinese)
    [41] Sod GA. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics, 1978, 27: 1-31 doi: 10.1016/0021-9991(78)90023-2
    [42] Lax PD. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics, 1954, 7: 159-193 doi: 10.1002/cpa.3160070112
    [43] Woodward P, Colella P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 1984, 54: 115-173 doi: 10.1016/0021-9991(84)90142-6
    [44] Shu CW, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes II. Journal of Computational Physics, 1989, 83: 32-78 doi: 10.1016/0021-9991(89)90222-2
    [45] Titarev VA, Toro EF. Finite-volume WENO schemes for three-dimensional conservation laws. Journal of Computational Physics, 2004, 201: 238-260 doi: 10.1016/j.jcp.2004.05.015
    [46] Toro EF, Titarev VA. TVD fluxes for the high-order ADER schemes. Journal of Scientific Computing, 2005, 24(3): 285-309 doi: 10.1007/s10915-004-4790-8
    [47] Titarev VA, Toro EF. WENO schemes based on upwind and centred TVD fluxes. Computers & Fluids, 2005, 34: 705-720
    [48] Schulz-Rinne CW, Collins JP, Harland MG. Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM Journal of Scientific Computing, 1993, 14(6): 1394-1414
    [49] Schulz-Rinne CW. Classification of the Riemann problem for two-dimensional gas dynamics. SIAM Journal of Mathematical Analysis, 1993, 24: 76-88 doi: 10.1137/0524006
    [50] Lax PD, Liu XD. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM Journal on Scientific Computing, 1998, 19(2): 319-340 doi: 10.1137/S1064827595291819
    [51] Vevek US, Zang B, New TH. Adaptive mapping for high order WENO methods. Journal of Computational Physics, 2019, 381: 162-188 doi: 10.1016/j.jcp.2018.12.034
    [52] Balsara D S, Rumpf T, Dumbser M, et al. Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. Journal of Computational Physics, 2009, 228: 2480-2516 doi: 10.1016/j.jcp.2008.12.003
    [53] Fan P, Shen Y, Tian B, et al. A new smoothness indicator for improving the weighted essentially non-oscillatory scheme. Journal of Computational Physics, 2014, 269: 329-354 doi: 10.1016/j.jcp.2014.03.032
    [54] Cockburn B, Shu CW. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics, 1998, 141: 199-224 doi: 10.1006/jcph.1998.5892
    [55] Zhu J, Qiu J. A new fifth order finite difference WENO scheme for solving hyperbolic conservation laws. Journal of Computational Physics, 2016, 318: 110-121 doi: 10.1016/j.jcp.2016.05.010
    [56] Shi J, Zhang YT, Shu CW. Resolution of high order WENO schemes for complicated flow structures. Journal of Computational Physics, 2003, 186(2): 690-696 doi: 10.1016/S0021-9991(03)00094-9
    [57] Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability. London: Oxford University Press, 1961
    [58] Frank A, Jones TW, Ryu D, Gaalaas J B. The magnetohydrodynamic Kelvin-Helmholtz instability: a two-dimensional numerical study. Astrophysics Journal, 1996, 460: 777-93 doi: 10.1086/177009
    [59] San O, Kara K. Evaluation of Riemann flux solvers for WENO reconstruction schemes: Kelvin-Helmholtz instability. Computers & Fluids, 2015, 117: 24-41
    [60] Kumar R, Chandrashekar P. Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws. Journal of Computational Physics, 2018, 375: 1059-1090 doi: 10.1016/j.jcp.2018.09.027
    [61] Hong Z, Ye ZY, Meng XZ. A mapping-function-free WENO-M scheme with low computational cost. Journal of Computational Physics, 2020, 405: 10914
  • 加载中
图(19) / 表(10)
计量
  • 文章访问数:  58
  • HTML全文浏览量:  16
  • PDF下载量:  8
  • 被引次数: 0
出版历程
  • 收稿日期:  2022-06-03
  • 录用日期:  2022-09-06
  • 网络出版日期:  2022-09-06

目录

    /

    返回文章
    返回