基于Hermite WENO格式的双曲守恒律高精度数值方法研究
High-Order Numerical Method Based on Hermite WENO Scheme for Hyperbolic Conservation Laws
摘要: 本文针对双曲守恒律方程，构造并系统研究了有限差分Hermite WENO (HWENO)格式，重点分析不同数值通量对格式精度、计算效率与间断分辨率的影响。在格式构造上，将数值通量分解为低阶通量与高阶修正项，采用HWENO重构得到半格点处的五阶精度近似，并对高分辨率项进行中心差分与迎风分裂处理，保证格式稳定性与收敛性。数值实验选取一维Euler方程的精度验证、Lax黎曼问题、Shu-Osher问题、双爆轰波相互作用等典型算例，对比LF、LLF、HLL、HLLC、FORCE、FLIC等通量的性能。结果表明，HWENO格式具备良好的无振荡特性与激波分辨能力；LF通量计算成本最低但耗散最大、精度最差，HLL与HLLC通量在精度、间断分辨率和计算耗时之间达到最优平衡。本文所构建的HWENO格式可为流体力学问题提供高效可靠的数值模拟工具。
Abstract: In this paper, a finite difference Hermite WENO (HWENO) scheme is constructed and systematically investigated for hyperbolic conservation laws, with a focus on analyzing the effects of different numerical fluxes on scheme accuracy, computational efficiency and discontinuity resolution. In the scheme construction, the numerical flux is decomposed into a low-order flux and a high-order correction term. Fifth-order accurate approximations at half-grid points are obtained via HWENO reconstruction, and central difference and upwind splitting are applied to high-resolution terms to ensure the stability and convergence of the scheme. Numerical experiments include typical cases such as accuracy test for one-dimensional Euler equations, Lax Riemann problem, Shu-Osher problem, double detonation wave interaction, where the performance of LF, LLF, HLL, HLLC, FORCE and FLIC fluxes is compared. The results demonstrate that the HWENO scheme exhibits excellent non-oscillatory behavior and shock-capturing capability. The LF flux has the lowest computational cost but the largest dissipation and the worst accuracy, while HLL and HLLC fluxes achieve the best balance among accuracy, discontinuity resolution and CPU time. The HWENO scheme developed in this paper can provide an efficient and reliable numerical simulation tool for problems in fluid mechanics.
文章引用：王泽如, 刘红霞. 基于Hermite WENO格式的双曲守恒律高精度数值方法研究[J]. 应用数学进展, 2026, 15(6): 311-323. https://doi.org/10.12677/aam.2026.156289

参考文献

[1] LeVeque, R.J. (2002) Finite Volume Methods for Hyperbolic Problems. Cambridge University Press. [Google Scholar] [CrossRef] 
[2] Toro, E.F. (2009) Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. 3rd Edition, Springer.
[3] Liu, X.D., Osher, S. and Chan, T. (1994) Weighted Essentially Non-Oscillatory Schemes. Journal of Computational Physics, 115, 200-212. [Google Scholar] [CrossRef] 
[4] Jiang, G.S. and Shu, C.W. (1996) Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics, 126, 202-228. [Google Scholar] [CrossRef] 
[5] Shu, C.W. (1998) Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. In: Cockburn, B., et al., Eds., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Springer, 325-432. [Google Scholar] [CrossRef] 
[6] Qiu, J.X. and Shu, C.W. (2004) Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method: One-Dimensional Case. Journal of Computational Physics, 193, 115-135. [Google Scholar] [CrossRef] 
[7] Qiu, J. and Shu, C. (2005) Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method II: Two Dimensional Case. Computers & Fluids, 34, 642-663. [Google Scholar] [CrossRef] 
[8] Zhao, Z. and Qiu, J.X. (2020) A Hermite WENO Method with Modified Ghost Fluid Method for Compressible Two-Medium Flow Problems. Communications in Computational Physics, 28, 743-773.
[9] Lax, P.D. (1954) Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation. Communications on Pure and Applied Mathematics, 7, 159-193. [Google Scholar] [CrossRef] 
[10] Harten, A., Lax, P.D. and Leer, B.V. (1983) On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws. SIAM Review, 25, 35-61. [Google Scholar] [CrossRef] 
[11] Toro, E.F., Spruce, M. and Speares, W. (1994) Restoration of the Contact Surface in the HLL-Riemann Solver. Shock Waves, 4, 25-34. [Google Scholar] [CrossRef] 
[12] Toro, E.F. and Titarev, V.A. (2006) MUSTA Fluxes for Systems of Conservation Laws. Journal of Computational Physics, 216, 403-429. [Google Scholar] [CrossRef] 
[13] Gottlieb, S. and Shu, C. (1998) Total Variation Diminishing Runge-Kutta Schemes. Mathematics of Computation, 67, 73-85. [Google Scholar] [CrossRef] 
[14] Shu, C.W. and Osher, S. (1989) Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes, II. Journal of Computational Physics, 83, 32-78. [Google Scholar] [CrossRef] 

为你推荐