完全守恒的半拉格朗日多矩约束有限体积方法研究

The traditional semi-Lagrangian method allows large CFL number, however, its numerical conservation can not be guaranteed since the point-wise upstream values of physical variables are obtained by numerical interpolation. In this project, a cellwisely high order, computationally efficient, locally and globally conserved semi-Lagrangian multi-moment finite volume method is presented through combining the multi-moment constrained finite volume method and semi-Lagrangian method. Based on the multi-moment constraint concep, high order reconstructions are conducted cell-wisely by making use of volume-integrated average (VIA), point value (PV) and derivative value (DV) if necessary. The evolution formulations are derived by adopting multi-moments as constraint conditions. The numerical conservation is rigorously guaranteed in the present method. This method maintains the advantage of semi-Lagrangian large CFL number and numerical accuracy, and the reconstruction technique within a single cell is much beneficial for massively parallel computation and due to substantial reduction in the data communication among the grid cells. It can probably be applied not only to tracer advection, but also to nonlinear dynamical model in atmospheric and ocean scopes. The numerical techniques presented in this project probably constitute a promising and practical numerical framework to develop atmospheric and oceanic general circulation models, and help to advance the atmospheric and oceanic models established by chinese scientists.

传统基于格点插值办法的半拉格朗日算法，其物理量守恒性不能够得到保证。 本项目采用多矩约束有限体积方法和半拉格朗日方法相结合的新思路，构建一种基于单网格的、高精度的、高效的、局地与全局完全守恒的半拉格朗日多矩有限体积算法。基于多矩约束概念，在单一计算网格内定义的体积积分平均值、点值以及导数值来重构高阶插值函数，通过多矩约束条件，得到多矩变量时间演变方程。采用体积积分值约束守恒概念可以保证了多物理量的严格数值守恒。本项目采用的方法预期不但能保持半拉格朗日大时间步长的计算优势，有利于改善模式计算精度，而且单网格重构技术有助于大型计算并行化处理，减少网格间数据通信量，其应用不仅仅局限于平流计算，还可以直接应用于复杂大气海洋非线性动力模式。本项目的数值技术预期能够构建一个实用、可行的大气海洋通用环流模式数值框架，有助于推进我国科学家研发的大气海洋模式系统的发展。

本项目针对传统基于格点插值办法的数值计算不守恒半拉格朗日算法，通过引入多矩约束的概念，发展了基于单网格的、高精度的、局地与全局完全守恒的半拉格朗日多矩有限体积算法。开展的研究内容有：（1）局地高阶重构技术；（2）局地与全局守恒技术；（3）单调限制器（limiter）方案技术；（4）多矩算法中点值与积分平均值之间的内在关系研究；（5）高效的半拉格朗日PV矩计算和数值模拟。通过4年的研究，本项目发展的半拉格朗日多矩约束有限体积算法，能够严格保证局地与全局数值守恒，解决了传统基于点值计算的一类半拉朗日方法不守恒性。发展守恒的半拉格朗日多矩有限体积算法能够高精度计算通量，并且易于扩展到多维非线性大气。本项目研制的方法不但能保持半拉格朗日大时间步长的计算优势，有利于改善模式计算精度，而且单网格重构技术有助于大型计算并行化处理，减少网格间数据通信量，在未来众核高性能计算中具有良好可扩展性的潜力。考虑到本项目研制的算法的点值计算具有高效性和灵活性，其应用不仅仅局限于平流计算，还可以直接应用于复杂大气海洋非线性动力模式。本项目的数值技术构建一个实用、可行的大气海洋通用环流模式数值框架，有助于推进我国科学家研发的大气海洋模式系统的发展。