论文标题
高级runge-kutta方法的边界处理方法
Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws
论文作者
论文摘要
在\ cite {zh2019}中，我们开发了一种用于隐式解释（IMEX）runge-kutta（RK）方法的边界处理方法，用于求解具有源项的双曲线系统。由于IMEX RK方法包括明确的特殊情况，因此这种边界处理方法自然也适用于显式方法。在本文中，我们研究了适用于双曲线保护定律的任意顺序的明确RK方案的情况。我们表明该方法不仅保留了显式RK方案的准确性，而且还具有良好的稳定性。这与\ cite {TS2010，TWSN2012}中的Lax-WendRoff方法相比有利，其中分析和数值实验先前已经验证了降级降低的存在\ cite \ cite {ts2010，twsn2012}。此外，我们证明了我们的方法在涉及负系数和下风空间离散的强稳定性（SSP）RK方案方面的性能很好。从数值上表明，当存在边界条件并使用所提出的边界处理时，与具有所有非负系数的方案相比，具有负系数的SSP RK方案仍然允许更大的时间步长。在这方面，我们的边界处理方法是对SSP RK方案的有效补充，具有/不为负系数的SSP RK方案对于双曲线保护定律的初始有限价值问题。
In \cite{ZH2019}, we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favourably to the inverse Lax-Wendroff method in \cite{TS2010,TWSN2012} where analysis and numerical experiments have previously verified the presence of order reduction \cite{TS2010,TWSN2012}. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all non-negative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.
