Robust and Efficient High Order Methods for Time Dependent Problems

鲁棒高效的高阶精度计算方法因其能够产生高保真度的模拟结果而在真实的世界问题的数值模拟中得到越来越多的应用。然而，这样的方法是不可用的或没有很好地理解流体动力学方程模拟高速流在航天器设计，燃烧，爆炸，天体物理射流，飓风，海啸，等离子体动力学，惯性约束聚变。该研究项目旨在开发用于模拟这些系统的改进的数值方法。 在设计鲁棒和更有效的高阶方法的进展将显着影响模拟技术的应用。 高阶精度数值方法在求解含时非线性问题中的应用还远远不能满足实际需要。与低阶方法相比，高阶方法更难稳定，并且由于更大的计算机存储成本，在实践中可能效率较低。因此，利用高精度方法求解真实的世界问题中的非线性流体力学方程仍然是一个挑战。本提案的目的是从具体角度解决这些现实世界的问题挑战。首先，人们希望通过保持物理量（如正性）的某些不变性来确保欧拉方案的鲁棒性。第二，人们希望设计更有效的实现非常高阶计划的复杂几何形状的曲线元素。

Robust and efficient high order accurate methods for computation have gained more and more popularity in the numerical modeling of real world problems for their ability to produce high fidelity simulations. However, such methods are not available or not well understood for hydrodynamics equations modeling high speed flows in spacecraft design, combustion, detonation, astrophysical jets, hurricanes, tsunamis, plasma dynamics, and inertial confinement fusion. This research project aims to develop improved numerical methods for simulation of these systems. Progress in designing robust and more efficient high order methods will significantly impact on simulation technology for such applications. The state of high order accurate numerical methods is still far from being practically satisfactory for time-dependent nonlinear problems. Compared to their low order counterparts, high order methods are much harder to stabilize and might be less efficient in practice due to much larger computer memory cost. Thus it remains challenging to utilize a high order accurate method to solve nonlinear hydrodynamics equations in real world problems. The objective of this proposal is to address these real-world problem challenges from specific perspectives. First of all, one would like to ensure the robustness of Eulerian schemes by preserving certain invariances of physical quantities such as positivity. Second, one would like to design more efficient implementations of very high order schemes on curved elements for complex geometries.