Abstract In this paper, high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are designed for solving hyperbolic conservation laws on triangular meshes. These multi-resolution WENO limiters are new extensions of the associated multi-resolution WENO finite volume schemes [49] , [50] which serve as limiters for RKDG methods from structured meshes [47] to triangular meshes. Such new WENO limiters use information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [24] , to build a sequence of hierarchical L 2 projection polynomials from zeroth degree to the highest degree of the RKDG method. The second-order, third-order, and fourth-order RKDG methods with associated multi-resolution WENO limiters are developed as examples, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong shocks or contact discontinuities by gradually degrading from the highest order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be any positive numbers on the condition that their sum equals one. This is the first time that a series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on triangular meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on triangular meshes. Extensive one-dimensional (run as two-dimensional problems on triangular meshes) and two-dimensional tests are performed to demonstrate the effectiveness of these RKDG methods with the new multi-resolution WENO limiters.

中文翻译：

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三角形网格上具有新型多分辨率 WENO 限制器的高阶 Runge-Kutta 不连续 Galerkin 方法

摘要 本文设计了具有多分辨率加权基本非振荡(WENO)限制器的高阶Runge-Kutta不连续伽辽金(RKDG)方法来求解三角形网格上的双曲守恒定律。这些多分辨率 WENO 限制器是相关多分辨率 WENO 有限体积方案 [49]、[50] 的新扩展，它们用作从结构化网格 [47] 到三角形网格的 RKDG 方法的限制器。这种新的 WENO 限制器基本上仅在问题单元本身内使用 DG 解的信息，该信息由原始 KXRCF 指标的新修改版本识别 [24]，以构建从零阶到最高阶的分层 L 2 投影多项式序列RKDG 方法的程度。二阶，三阶，并以具有相关多分辨率 WENO 限制器的四阶 RKDG 方法为例，该方法可以在平滑区域保持原始精度，并通过从最高阶逐渐退化到强冲击或接触不连续性附近保持本质上的非振荡特性。第一个订单。新的多分辨率 WENO 限制器程序内的线性权重可以是任何正数，条件是它们的总和等于 1。这是第一次以 WENO 方式应用故障单元本身内的一系列不同次数的多项式来修改三角形网格上故障单元中的 DG 解。这些新的 WENO 限制器构建起来非常简单，并且可以在非结构化网格上轻松实现任意高阶精度和更高维度。这种空间重建方法提高了在三角形网格上原始 DG 方法的相同紧凑空间模板上模拟的鲁棒性。进行了广泛的一维（在三角形网格上作为二维问题运行）和二维测试，以证明这些 RKDG 方法与新的多分辨率 WENO 限制器的有效性。