Runge-Kutta间断Galerkin方法的各向异性自适应方法及其应用

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.

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。

本项目以h自适应RKDG（Runge-Kutta间断Galerkin）方法为基础，研究各向异性的h自适应RKDG方法以及此方法的推广应用，通过对自适应引入各向异性，优化并提高其自适应效率，使得用新方法求解这些适用目标方程时能节省数值模拟计算成本，提高数值求解效率。项目组在项目实施期内投入了很多时间和精力，认真执行项目计划，开展研究工作。项目组所取得的研究成果已经发表5篇学术论文，其中3篇被SCI收录。对比项目研究计划，项目组基本完成了项目的研究内容，实现了预期的研究目标。