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arXiv 2008.11981math.NAcs.NA
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双曲型守恒律间断Galerkin方法通量和斜率限制的新观点

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

Dmitri Kuzmin

摘要:

在这项工作中,我们讨论和发展多维限制技术的不连续伽辽金(DG)离散的标量双曲问题。为了确保每个单元平均值满足局部离散最大值原理(P1),我们施加不等式约束的局部Lax-Friedrichs通量的分段线性(P1)近似。由于分段常数(P0)的版本对应于一个保持性能的低阶有限体积法,有效性的条件下,总是可以强制使用斜率和/或通量限制器。我们表明,(目前相当罕见)使用直接通量限制,使人们有可能构建更准确的DMP满意的近似,其中一个弱形式的斜率限制,以防止无界增长的解决方案梯度。此外,通量和斜率都可以限制的方式,即使在稳定状态下,也会产生具有明确残差的非线性问题。我们推导/目前的斜率限制器的基础上不同种类的不等式约束,讨论其性质,并引入新的各向异性限制器的问题,需要不同的处理不同的空间方向。在通量限制阶段,可以通过使用局部边界的自定义来考虑问题的各向异性。在斜率限制阶段,我们使用来自单元平均值的低阶重建来调整各个方向导数的大小以定义边界。通过这种方式,我们避免了在光滑峰和内部/边界层中对解析良好的导数进行不必要的限制。所选算法的性能进行了探讨,在数值研究的DG-P1离散化的二维测试问题。在Hp自适应DG方法的上下文中,新的限制程序可以用于标记为“困扰”的平滑度指标的宏单元的P1子单元中。

英文摘要:

In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a local discrete maximum principle (DMP), we impose inequality constraints on the local Lax-Friedrichs fluxes of a piecewise-linear (P1) approximation. Since the piecewise-constant (P0) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. Moreover, both fluxes and slopes can be limited in a manner which produces nonlinear problems with well-defined residuals even at steady state. We derive/present slope limiters based on different kinds of inequality constraints, discuss their properties and introduce new anisotropic limiters for problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P1 subcells of macroelements marked as `troubled' by a smoothness indicator.

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