High-order adaptive methods for hyperbolic conservation laws

本文提出了一种新的高阶自适应算法，用于使用不连续伽辽金方法（DGM）求解双曲型守恒律的数值解。这些方法对于用偏微分方程建模的大规模问题的解的计算变得越来越重要。它们具有元素守恒性、高阶精度、易于自适应和简单的并行求解过程等特点。它们被用于流体动力学、空气声学、电磁学和光学等不同的应用中。这项工作将导致更准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能够计算出高度精确的数值解。为了使这种方法具有竞争力，它必须高效而迅速地做到这一点。本提案将朝着这个方向开发若干工具。自适应算法将计算资源集中在最需要的区域。因此，需要更少的细胞来达到规定的精度。新的自适应算法能够结合两种类型的自适应：h-细化（网格充实）和p-细化（顺序变化）将被开发。对于这种完全自适应的方案，将研究新的稳定（限制）技术。我们将通过开发具有显式龙格-库塔时间积分器的直线方法的时间自适应方案来进一步寻求效率。通过在每个单元格上使用本地时间步长，该方案将需要更少的函数计算次数。因此，仿真将在更短的时间内完成。然后，我们将开发具有嵌入边界的笛卡尔网格的不连续伽辽金方法。这样的网格比非结构化的三角形/四面体网格更容易构建，计算速度更快。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.