High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes
High-order Runge-Kutta discontinuous Galerkin methods with a new type of multi-resolution WENO limiters on triangular meshes
复制标题
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
DOI：
10.1016/j.apnum.2020.03.013
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发表时间：
2020
影响因子：
2.8
通讯作者：
Qiu Jianxian
中科院分区：
文献类型：
--
作者：
Zhu Jun;Shu Chi-Wang;Qiu Jianxian
在本文中，设计了具有多分辨率加权本质非振荡（WENO）限制器的高阶龙格-库塔不连续伽辽金（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，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 限制器基本上仅使用问题单元本身内的 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 RKDG 方法从零次到最高次的分层 L 2 投影多项式序列。作为例子，开发了具有相关多分辨率WENO限制器的二阶、三阶和四阶RKDG方法，该方法可以通过从最高阶逐渐降级到一阶来保持平滑区域中的原始精度，并在强冲击或接触不连续性附近保持基本的非振荡特性。新的多分辨率WENO限制器过程中的线性权重可以是任何正数，只要它们的总和等于1。这是第一次以 WENO 方式应用问题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问题单元中的 DG 解。这些新的 WENO 限制器构造起来非常简单，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的原始 DG 方法的相同紧凑空间模板上进行模拟的鲁棒性。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。
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.