Discontinuous Galerkin Methods for Solution of Hyperbolic Conservation Laws on Cartesian Grids

该研究计划的目的是开发新的高阶数值技术的解决方案的双曲守恒律的框架内的间断Galerkin方法（DGM）。该方法将应用于三维计算机模拟的可压缩流体流动，太赫兹集成天线，和风乐器。 我们将把我们在嵌入几何的笛卡尔网格上的工作扩展到三维空间。由于笛卡尔网格的规则结构，笛卡尔网格方法简化并加速了计算域内的计算。然而，当具有复杂几何特征的物理对象从网格中切割出来时，所创建的小且不规则的单元难以进行数值处理。如何应对这些挑战将是本研究的重点。 我们还将致力于DG方法的分析，并将最近提出的修改DGM高维问题。我们将分析色散和耗散误差，线性稳定性和超收敛之间的关系。我们将致力于能够提取数值解的超收敛分量的后处理技术，并旨在提高收敛速度。 该提案的一个重要部分是软件开发。为了能够应用新技术，我们将开发和实现一个三维笛卡尔网格生成器，能够创建复杂的，适应的几何网格。我们还将使用新的计算机架构，如图形处理单元（GPU）。在当前中等成本的视频卡上的计算可以比在传统CPU上快几个数量级。这将使我们能够将高保真计算从高性能集群转移到桌面工作站。这反过来又可以使真实的时间计算成为可能，并导致使用数值模拟作为拟议类型应用的建模工具。

The aim of the proposed research program is to develop novel high-order numerical techniques for solution of hyperbolic conservation laws in the framework of the discontinuous Galerkin method (DGM). The method will be applied to three-dimensional computer simulations of compressible fluid flows, terahertz integrated antennas, and wind musical instruments. We will extend our work on Cartesian grids with embedded geometries to three-dimensional space. Cartesian grid methods simplify and speed-up computations inside of the computational domain due to the regular structure of Cartesian grids. However, small and irregular cells created when a physical object with complex geometrical features is cut out of the grid are difficult to handle numerically. Finding solutions to these challenges will be the focus of this research. We will also work on analysis of the DG method and extend a recently proposed modified DGM to higher dimensional problems. We will analyse the relation between dispersion and dissipation errors, linear stability, and superconvergence. We will work on postprocessing techniques capable of extracting superconverging components of the numerical solution and aiming to increase the rate of convergence. An important part of this proposal is software development. In order to be able to apply the new techniques, we will develop and implement a three dimensional Cartesian mesh generator capable of creating complex, adapted to the geometry grids. We will also use novel computer architectures such as Graphics Processing Units (GPUs). Computations on current medium cost video cards can be performed orders of magnitude faster than on traditional CPUs. This will allow us to move high fidelity computations from high-performance clusters to desktop workstations. This, in turn, could make real time computations possible and lead to the use of numerical simulations as modelling tools for the proposed types of applications.