There is a pressing demand for robust, high-order baseline schemes for conservation laws that minimize reliance on supplementary stabilization. In this work, we respond to this demand by developing new baseline schemes within a nodal discontinuous Galerkin (DG) framework, utilizing upwind summation-by-parts (USBP) operators and flux vector splittings. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our method encompasses a broader class of USBP operators, not limited to equidistant grid points. This approach facilitates the development of novel USBP operators on Legendre--Gauss--Lobatto (LGL) points, which are suited for nodal discontinuous Galerkin (DG) methods. The resulting DG-USBP operators combine the strengths of traditional summation-by-parts (SBP) schemes with the benefits of upwind discretizations, including inherent dissipation mechanisms. Through numerical experiments, ranging from one-dimensional convergence tests to multi-dimensional curvilinear and under-resolved flow simulations, we find that DG-USBP operators, when integrated with flux vector splitting methods, foster more robust baseline schemes without excessive artificial dissipation.

翻译：当前迫切需要构建鲁棒的高阶基准格式用于守恒律，以最小化对额外稳定化措施的依赖。在本工作中，我们通过基于节点间断伽辽金（DG）框架，利用迎风求和分部（USBP）算子和通量矢量分裂技术，开发了新的基准格式以响应这一需求。为此，我们证明了在任意网格点上USBP算子的存在性，并提供了其构造的简明流程。我们的方法涵盖了一类更广泛的USBP算子，不局限于等距网格点。该方法促进了在适用于节点间断伽辽金（DG）方法的Legendre--Gauss--Lobatto（LGL）点上构建新型USBP算子。由此得到的DG-USBP算子融合了传统求和分部（SBP）格式的优势与迎风离散化的优点，包括固有的耗散机制。通过从一维收敛性测试到多维曲线坐标及欠分辨流动模拟的一系列数值实验，我们发现，当DG-USBP算子与通量矢量分裂方法结合时，能够形成更鲁棒的基准格式，而无需引入过度的数值耗散。