High-order adaptive methods for hyperbolic conservation laws

这一建议旨在发展新的高阶自适应算法，用于双曲型守恒律数值解的间断Galerkin方法(DGM)。这些方法正在成为计算以偏微分方程组为模型的大规模问题的重要工具。它们吸引人的特点是单元守恒性、高精度、易适应性和简单的并行求解过程。它们被广泛应用于流体动力学、空气声学、电磁学和光学等领域。这项工作将导致更精确的计算机模拟和更短的设计时间，通过用高阶基表示解，DGM能够计算出高精度的数值解。为了使这种方法具有竞争力，它必须高效而快速地做到这一点。这项提议将朝着这个方向发展几个工具。自适应算法将计算资源集中在最需要它们的区域。因此，需要较少的单元即可达到规定的精度。将开发能够结合两种类型的自适应的新的自适应算法：H-精化(网格浓缩)和P-精化(阶变)。对于这种完全自适应的方案，将研究新的稳定(限制)技术。对于具有显式Runge-Kutta时间积分器的线法，我们将进一步开发一种时间自适应的方法来寻求效率。通过在每个单元格上使用本地时间步长，该方案需要的函数求值的总体数量将更少。因此，模拟将在更短的时间内完成。然后，我们将在具有嵌入边界的笛卡尔网格上发展一种不连续Galerkin方法。这种网格比非结构化三角形/四面体网格更容易构造，计算速度也更快。研究生和本科生的培训将是拟议计划的重要组成部分。

This proposal is aimed at the development of new high-order adaptive algorithms for the numerical solution of hyperbolic conservation laws using discontinuous Galerkin methods (DGM). These methods are becoming important  for the computation of  solutions of large scale problems modeled by partial differential equations. Their attractive features are an element-wise conservation property, high-order accuracy, easy adaptivity, and simple parallel solution procedures. They are used in such diverse applications as fluid dynamics, aeroacoustics, electromagnetics, and optics. This work will lead to more accurate computer simulations and shorter design times.By representing the solution via a high-order basis, the DGM is capable of computing a highly accurate numerical solution. For the method to be competitive, it must do so efficiently and rapidly. This proposal will develop several tools in that direction. Adaptive algorithms concentrate computing resources in regions where they are most needed. Thus, fewer cells are needed to achieve the prescribed accuracy. New adaptive algorithms capable of combining two types of adaptivity: h-refinement (mesh enrichment) and p-refinement (order variation) will be developed.  New stabilizing (limiting) techniques will be investigated for such fully adaptive schemes. We will further seek efficiency by developing a time-adaptive scheme for the method of lines with explicit Runge-Kutta time integrators. By using a local time step on each cell, the scheme will need an overall fewer number of function evaluations. Thus, the simulation will be completed in a shorter time. We then will develop a discontinuous Galerkin method on Cartesian grids with embedded boundaries. Such grids are easier to construct and faster to compute on than unstructured triangular/ tetrahedral meshes. Training of graduate and undergraduate students will be an important component of the proposed program.