基于静动态混合重构的DG/FV混合格式

通过比较紧致格式和间断Galerkin(DG)格式,提出了"静态重构"和"动态重构"的概念,对有限体积方法和DG有限元方法进行统一的表述.借鉴有限体积的思想,发展了基于"混合重构"技术的一类新的DG格式,称之为间断Galerkin有限元/有限体积混合格式(DG/FV格式).该类混合格式通过适当地扩展模板(拓展至紧邻单元)重构单元内的高阶多项式分布,在提高精度的同时,减少了传统DG格式的计算量和存储量.通过典型一维和二维标量方程的计算发现新的混合格式在有些情况下具有超收敛(superconvergence)性质.

A CLASS OF DISCONTINUOUS GALERKIN/FINITE VOLUME HYBRID SCHEMES BASED ON THE "STATIC RE-CONSTRUCTION" AND "DYNAMIC RE-CONSTRUCTION"

By comparing the compact finite difference schemes and discontinuous Galerkin (DG) methods, the concepts of ``static re-construction' and ``dynamic re-construction' are proposed for high-order numerical schemes. Based on the new concept of ``hybrid re-construction', a novel class of DG/finite volume hybrid schemes (DG/FV schemes) is presented. In our DG/FV schemes, the lower-order derivatives are computed locally in a cell by traditional DG schemes (called as ``dynamic re-construction'), while the higher-order derivatives are constructed by the ``static re-construction' of finite volume schemes, using the known lower-order derivatives in the cell itself and in the neighbor cells. The DG/FV hybrid schemes can reduce the CPU time and storage memory greatly than the traditional DG schemes with the same order of accuracy, and can be extended directly for unstructured and hybrid grids as the DG and/or FV methods. The DG/FV hybrid schemes are applied for 1D and 2D scalar conservation law. The numerical results demonstrate the accuracy, the efficiency, and the super-convergence property in our third-order DG/FV hybrid schemes.