基于HWENO格式的不同数值通量性能对比研究
Performance Comparison of Different Numerical Fluxes Based on HWENO Schemes
摘要: 为对比不同数值通量与HWENO格式结合的性能，本文以一维及二维双曲守恒律方程为研究对象，构造了基于多种典型数值通量的有限差分HWENO格式，阐述了格式构造流程与各通量实现细节。通过多个经典算例，从收敛阶数、计算效率与间断分辨率三方面定量对比了不同通量下的格式表现。结果表明，Godunov和EO通量格式间断分辨率最优但计算成本较高，而LF通量格式成本最低但数值耗散最大。本文研究为双曲守恒律高阶数值方法的通量选取提供了参考。
Abstract: To compare the performance of HWENO schemes combined with different numerical fluxes, this paper takes one- and two-dimensional hyperbolic conservation laws as the research objects, constructs finite difference HWENO schemes based on several typical numerical fluxes, and elaborates the construction process of the schemes and the implementation details of each flux. Through several classical numerical examples, the performance of the schemes with different fluxes is quantitatively compared from three aspects: convergence order, computational efficiency and discontinuity resolution. The results show that the HWENO schemes with Godunov and EO fluxes have the best discontinuity resolution but high computational cost, while the LF flux scheme has the lowest cost but the largest numerical dissipation. This study provides a reference for the selection of numerical fluxes in high-order numerical methods for hyperbolic conservation laws.
参考文献
|
[1]
|
Liu, H. and Qiu, J. (2016) Finite Difference Hermite WENO Schemes for Conservation Laws, II: An Alternative Approach. Journal of Scientific Computing, 66, 598-624. [Google Scholar] [CrossRef]
|
|
[2]
|
Qiu, J., Khoo, B.C. and Shu, C. (2006) A Numerical Study for the Performance of the Runge-Kutta Discontinuous Galerkin Method Based on Different Numerical Fluxes. Journal of Computational Physics, 212, 540-565. [Google Scholar] [CrossRef]
|
|
[3]
|
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]
|
|
[4]
|
Shu, C. and Osher, S. (1988) Efficient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes. Journal of Computational Physics, 77, 439-471. [Google Scholar] [CrossRef]
|
|
[5]
|
Titarev, V.A. and Toro, E.F. (2002) ADER: Arbitrary High Order Godunov Approach. Journal of Scientific Computing, 17, 609-618. [Google Scholar] [CrossRef]
|
|
[6]
|
Toro, E.F. (1997) Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer. [Google Scholar] [CrossRef]
|
|
[7]
|
Toro, E.F. (2001) Shock Capturing Methods for Free Surface Shallow Flows. Wiley.
|
|
[8]
|
Toro, E.F. (2003) Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Isaac Newton Institute for Mathematical Sciences, University of Cambridge.
|
|
[9]
|
Osher, S. and Solomon, F. (1982) Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws. Mathematics of Computation, 38, 339-374. [Google Scholar] [CrossRef]
|
|
[10]
|
Pirozzoli, S. (2002) Conservative Hybrid Compact-Weno Schemes for Shock-Turbulence Interaction. Journal of Computational Physics, 178, 81-117. [Google Scholar] [CrossRef]
|