Positivity preserving limiter and new development on elliptic interface problems

这项研究将有助于从根本上理解在车辆设计和航空航天工程中广泛应用的算法开发。PI和她的合作者将对算法的性能和特性进行数学研究。具体来说，所提出的方法有助于稳定算法的实现，从而可以进行物理性质的数值研究，例如快速飞行的飞机周围的压力分布。与实验室实验相比，计算机模拟成本更低，效率更高。在这个项目中，PI还开发了一个新的思想和新的算法来解决接口问题。一个应用是复合材料研究，其中需要高精度和高效的求解器。研究者将把研究与教育活动结合起来，并在更广泛的背景下交流研究。对于一般的对流扩散方程，PI将发展出满足极限的三阶极大值原理。目的是证明高阶多项式解保持在给定边界内而不失去精度。进一步推广了可压缩Navier-Stokes方程保正极限的研究。在这个项目中，PI还开发了新的方法来解决椭圆界面问题，网格要么对准界面，要么穿过界面。该研究基于PI先前设计的直接不连续伽辽金方法。凭借在数值通量公式上额外的灵活性，PI能够在非结构三角形网格上以至少三阶精度证明满足严格极大值原理的二次多项式数值解。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在所有时间水平上都保持为正。作为一个副产品，多项式解的边界或保持解的正性可以被认为是一个强稳定性结果。本研究结果将提高数值方法解决计算流体力学难题的能力。对于椭圆界面问题，PI将修改在单元边缘定义的数值通量，以隐式地强制执行界面解跳和通量跳条件。

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.