Positivity preserving limiter and new development on elliptic interface problems

该研究将有助于对广泛应用于飞行器设计和航空航天工程的算法开发的基本理解。PI及其合作者将对算法的性能和属性进行数学研究。具体而言，所提出的方法有助于稳定的算法实现，例如，可以进行物理特性的数值研究，例如，快速飞行的飞机周围的压力分布。与实验室实验相比，计算机模拟更便宜，更有效。在这个项目中，PI还开发了一个新的想法和新的算法接口问题。一个应用是复合材料的研究，其中需要高精度和高效率的求解器。调查员将把研究与教育活动结合起来，并在更广泛的范围内传播研究成果。 PI将发展一般对流扩散方程的三阶极限值原理。目的是证明高阶多项式解保持在给定的范围内而不损失精度。PI进一步扩展了可压缩Navier-Stokes方程的保正极限。在这个项目中，PI还开发了新的方法来解决椭圆界面问题，网格可以对齐界面或穿过界面。该研究是基于PI先前设计的直接间断Galerkin方法。由于数值通量公式具有额外的灵活性，PI设法证明了非结构三角形网格上满足严格最大值原理的二次多项式数值解至少具有三阶精度。对网格没有几何限制，允许使用钝角三角形。PI将证明可压缩Navier-Stokes方程的密度和压力近似值在所有时间水平上都保持为正值。作为副产品，约束多项式解或保持解的正性可以被认为是强稳定性结果。本文的研究成果将有助于提高数值方法解决计算流体力学中具有挑战性问题的能力。对于椭圆界面问题，PI将修改在单元边缘定义的数值通量，以隐含地强制执行界面解跳跃和通量跳跃条件。

This research will contribute to the fundamental understanding of algorithm development that are widely applied in vehicles designs and aerospace engineering. The PI and her collaborators will perform mathematical studies on the performance and properties of the algorithms. Specifically the proposed method assists to stabilize the algorithm implementation such that numerical studies of physical properties, for example the pressure distributions around fast flying airplanes, can be carried out. Comparing with lab experiments, computer simulations are way less expensive and more efficient. In this project, the PI also develops a new idea and new algorithms for interface problems. One application is the compound materials studies in which highly accurate and efficient solvers are demanded. The investigator will integrate research with education activities and communicate the research in a broader context. The PI will develop third order maximum principle satisfying limiter for general convection diffusion equations. The objective is to prove the high order polynomial solutions staying in the given bounds without losing accuracy. The PI further extends the studies to obtain positivity preserving limiter for compressible Navier-Stokes equations. In this project, the PI also develops new methods to solve elliptic interface problems with mesh either aligned to or cut through the interface. The research is based on the direct discontinuous Galerkin methods previously designed by the PI. With the extra flexibility on the numerical flux formula, the PI manages to prove the quadratic polynomial numerical solutions satisfying strict maximum principle on unstructured triangular meshes with at least third order of accuracy. There is no geometric restriction on the meshes and obtuse triangles are allowed. The PI will prove the density and pressure approximations to compressible Navier-Stokes equations being maintained positive at all time levels. As a by-product, bounding the polynomial solutions or preserving the solution's positivity can be considered as a strong stability result. The findings of this research will improve the capability of a numerical method to those challenging problems from computational fluid dynamics. For elliptic interface problems, the PI will modify the numerical fluxes defined at element edges to implicitly enforce the interface solution jump and flux jump conditions.