间断Galerkin有限元方法是一类用完全不连续基函数的有限元方法，它拥有局部守恒、高阶精度、容易实现并行化，适用于复杂区域的计算等优点。本项目将主要针对双曲守恒律含有激波等间断的问题，设计既能保持高阶精度又能抑制间断处非物理振荡的紧致限制器，并且推广紧致限制器的应用，更准确地模拟含有激波、接触间断、大梯度、大变形等间断现象的问题及类似问题；本项目也将针对等离子体物理的基本模型Vlasov系统设计能量守恒算法；并将利用傅里叶分析对间断Galerkin有限元方法进行误差估计，研究各种间断Galerkin格式在特定点的超收敛性质。

Discontinuous Galerkin methods are a class of finite element methods using completely discontinuous basis functions. They enjoy several advantages, such as local conservation, high order accuracy, excellent parallel efficiency, the allowance of arbitrary triangulation and etc. This project will mainly focus on designing compact limiters for hyperbolic conservation laws containing shocks and other discontinuities, with the goal of obtaining the high order accuracy of the original discontinuous Galerkin method and non-oscillatory property near discontinuities, and applying the compact limiters to various problems that contain shocks, contact discontinuities and other discontinuities. This project is also devoted to designing energy-conserving methods for Vlasov systems, one of the fundamental models in plasma physics. Further more, this project will study the error estimate of discontinuous Galerkin methods using Fourier analysis and study the superconvergence property of various DG methods at some points.