High order upwind summation-by-parts finite difference operators have recently been developed. When combining with the simultaneous-approximation-term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

翻译：近年来，发展了高阶迎风格式求和分部有限差分算子。结合同时近似项方法施加边界条件时，该方法的收敛速度优于传统求和分部算子。我们通过标准模态分析，针对一类双曲型偏微分方程证明了此类方法的收敛阶。分析表明，施加边界条件的罚参数会影响稳定方法的收敛速率。此外，为求解具有间断数据的问题，我们将该方法扩展至具备加权本质无振荡特性。整体方法稳定，在光滑问题上能达到高阶精度，并能求解含间断的问题。