Robust and Efficient High Order Methods for Time Dependent Problems

鲁棒高效的高阶精确计算方法因其能够产生高保真的模拟结果而在现实世界问题的数值模拟中得到越来越多的应用。然而，对于航天器设计、燃烧、爆炸、天体物理射流、飓风、海啸、等离子体动力学和惯性约束聚变等高速流动的流体动力学方程建模，这些方法是不可用的，或者不是很好理解。本研究项目旨在开发改进的数值方法来模拟这些系统。设计鲁棒和更高效的高阶方法的进展将对此类应用的仿真技术产生重大影响。对于时变非线性问题，高阶精确数值方法的现状还远远不能令人满意。与低阶方法相比，高阶方法更难以稳定，并且由于更大的计算机内存成本，在实践中可能效率较低。因此，在实际问题中，利用高阶精确方法求解非线性流体力学方程仍然具有挑战性。本提案的目的是从具体的角度解决这些现实世界的问题挑战。首先，人们希望通过保留某些物理量的不变性（如正数）来确保欧拉格式的鲁棒性。其次，人们想要设计更有效的实现非常高阶方案在复杂几何形状的弯曲元素。

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.