Positivity preserving limiter and new development on elliptic interface problems

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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.