自适应Runge-Kutta间断Galerkin（RKDG）方法兼有RKDG方法精度高、求解高效、易于处理复杂计算区域和边界条件、并行效率高等优点以及自适应方法节约计算成本、提高求解效率等优点，是当今科学计算高分辨方法的研究前沿与热点。多维问题的数值解在局部，甚至整体上，不同方向的变化或性质可以有很大的区别。而一般的h自适应方法对所有方向统一处理，同时进行加密或合并，导致自适应效率低下，计算成本升高。本项目研究RKDG方法的各向异性自适应方法，及其向Hamilton-Jacobi方程和局部间断Galerkin方法的推广应用，目的是优化提高原自适应方法的自适应效率，使它们在求解相应方程时，能节省更多的数值模拟计算成本，进一步提高数值求解效率。

Runge-Kutta discontinuous Galerkin (RKDG) methods are high-order accurate, highly efficient and highly parallelizable methods which can easily handle complicated geometries, boundary conditions and hp-adaptivity. Adaptive techniques are widely used for their capability of saving the computational cost and improving the efficiency of numerical simulations. Adaptive RKDG methods have both advantages of RKDG methods and adaptive methods and are now the focus and forefront of research in scientific computing. Numerical solutions of multidimensional problems have different properties in different dimensions, locally, or sometimes even globally. Most of h-adaptive methods refine and coarsen mesh in all dimensions simultaneously, which decreases the efficiency of adaptivity and increases the computational cost. This project works on anisotropic adaptive RKDG methods and their extensions to Hamilton-Jacobi equations and local discontinuous Galerkin methods with an objective of improving the efficiency of former adaptive methods and saving more computational cost.