中文摘要：

高精度数值方法在求解对流扩散问题时一般会在解的间断处引入非物理数值振荡并导致求解不稳定。为了抑制间断解附近的非物理振荡，本文开发了一种适用于任意网格的紧致限制器。该限制器通过局部线性重构估计数值解的局部最大值和最小值，并将单元节点值限制在该范围内，抑制非物理振荡并保证数值稳定性。通过现今常用的分层限制思想，可将该限制器推广到任意高次多项式近似情况。文中基于间断有限元方法对该限制器进行了阐述，并通过一些典型算例对该限制器进行了验证，结果表明，该限制器具有紧致、保持空间精度和有效抑制非物理振荡的优点。

英文摘要：

Use of high-order methods to convection-dominated transport problems tends to cause spurious oscillations and nonlinear instability in the approximate solution. To prevent the spurious oscillations, an improvement of the vertex-based hierarchical slope limiter was proposed in this paper. Local linear reconstruction is used to estimate upper and lower bounds for the values at all vertices of each element with only data of those neighbors with a common edge with the concerned element required to keep compactness of the scheme. For the higher-order approximate solutions, the hierarchical limiting strategy is used with the same compactness kept. The supposed improved vertex-based limiter is used with a discontinuous Galerkin method to several problems to verify its properties. The results show that the limiter can keep lower numerical dissipation, prevent spurious-oscillations while keeping compact.