Abstract In this paper, we investigate designing a new type of high-order finite difference multi-resolution trigonometric weighted essentially non-oscillatory (TWENO) schemes for solving hyperbolic conservation laws and some benchmark highly oscillatory problems. We only use the information defined on a hierarchy of nested central spatial stencils in a trigonometric polynomial reconstruction framework without introducing any equivalent multi-resolution representations. These new finite difference trigonometric WENO schemes use the same large stencils as the classical WENO schemes (Jiang and Wu, 1996; Shu, 2009), could obtain the optimal order of accuracy in smooth regions, and simultaneously suppress spurious oscillations near strong discontinuities. The linear weights of such multi-resolution trigonometric WENO schemes can be any positive numbers on condition that their summation is one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference trigonometric WENO schemes. These new trigonometric WENO schemes are simple to construct and can be easily implemented to arbitrary high-order accuracy in multi-dimensions. Some benchmark examples including some highly oscillatory problems are given to demonstrate the robustness and good performance of these new trigonometric WENO schemes.

中文翻译：

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一种用于双曲守恒律和高振荡问题的新型高阶多分辨率三角 WENO 方案

摘要 在本文中，我们研究了设计一种新型的高阶有限差分多分辨率三角加权本质非振荡 (TWENO) 方案，以解决双曲守恒定律和一些基准高振荡问题。我们仅在三角多项式重建框架中使用在嵌套中央空间模板的层次结构上定义的信息，而没有引入任何等效的多分辨率表示。这些新的有限差分三角 WENO 方案使用与经典 WENO 方案相同的大模板（Jiang 和 Wu，1996；Shu，2009），可以获得平滑区域的最佳精度顺序，同时抑制强不连续性附近的虚假振荡。这种多分辨率三角 WENO 方案的线性权重可以是任何正数，条件是它们的和为 1。这是第一次将一系列大小不等的分层中心空间模板用于设计高阶有限差分三角 WENO 方案。这些新的三角 WENO 方案构造简单，可以轻松实现多维任意高阶精度。给出了一些基准示例，包括一些高振荡问题，以证明这些新三角 WENO 方案的鲁棒性和良好性能。这些新的三角 WENO 方案构造简单，可以轻松实现多维任意高阶精度。给出了一些基准示例，包括一些高振荡问题，以证明这些新三角 WENO 方案的鲁棒性和良好性能。这些新的三角 WENO 方案构造简单，可以轻松实现多维任意高阶精度。给出了一些基准示例，包括一些高振荡问题，以证明这些新三角 WENO 方案的鲁棒性和良好性能。