Efficient High-Order Methods for Solution of Hyperbolic Problems

我们周围的物理过程，例如天气锋的运动，可以使用称为偏微分方程的数学对象来描述。这些方程也被工程师用于开发新产品或改进现有产品;例如，它们可以用来计算飞机机翼的升力和阻力。这些方程太复杂，无法精确求解，因此我们采用数值方法近似求解。 该提案的目的是开发，分析和实施的数值方法称为间断伽辽金（DG）方法。 我们建议在应用于双曲守恒律时对DG方法有更好的数学理解，并使其更加有效和鲁棒。 DG方法在正确的条件下是高度准确的，但可能需要比其他方法更多的计算资源。我们试图理解为什么它是如此准确，通过分析其色散和耗散特性的非结构化计算网格在二维和三维空间。有了这种理解，我们将得出新的方法，可能有一个更好的权衡成本和准确性方面，特别是当应用到问题，发展冲击波。 我们还将致力于该计划的组成部分，如新的稳定技术，以及误差估计和自适应技术。这些都需要使该方法更有效，更适合于实际应用。为了产生精确的解并以最有利的方式分配计算资源，需要估计该方案所犯的误差。稳定化技术被用来抑制非物理振荡的数值解附近的间断。缺乏稳健的方案稳定方法是阻止方案充分发挥其潜力的主要问题之一。 这一建议的预期结果是数值分析领域的进步。我们还希望，开发的算法和方法将导致更快，更有效的计算与应用程序的可压缩流问题。

Physical processes around us, for example the movement of weather fronts, can be described using mathematical objects called partial differential equations. These equations are also used by engineers in developing new products or improving existing ones; for example they can be used to calculate lift and drag on an airplane wing. Such equations are too complex to be solved exactly, so we employ numerical methods to solve them approximately. The aim of this proposal is the development, analysis and implementation of the numerical method called the discontinuous Galerkin (DG) method. We propose to work on a better mathematical understanding of the DG method when applied to hyperbolic conservation laws and to make it more efficient and robust. The DG method is highly accurate in right conditions but might require more computational resources than other methods. We seek to understand why exactly it is so much more accurate by analyzing its dispersive and dissipative properties on unstructured computational meshes in two- and three-dimensional spaces. With this understanding, we will derive new methods that might have a better trade-off in terms of cost and accuracy, especially when applied to problems that develop shock waves. We will also work on components of the scheme such as new stabilization techniques, as well as error estimation and adaptive techniques. These are needed to make the method more efficient and more suitable for practical applications. Estimation of the error that the scheme commits is needed in order to produce an accurate solution and to distribute computational resources in the most advantageous way. Stabilization techniques are used to suppress non-physical oscillations in the numerical solutions near discontinuities. Absence of a robust method of scheme stabilization is one of the main issues to making preventing the scheme from attaining its full potential. The expected result of this proposal is the advancement of the field of numerical analysis. We also expect that the developed algorithms and approaches will lead to faster and more efficient computations with applications to compressible flow problems.