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

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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.