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 Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
 Speaker: Prof. Qinghai Zhang，School of Mathematical Sciences, Zhejiang University Inviter: 鄭偉英 研究員 Title: A Fourth-order Cut-cell Method for Elliptic and Parabolic Problems on Irregular Domains with Structured Rectangular Grids Time & Venue: 2019.11.14 14:00-15:00 N212 Abstract: Classical ?nite-di?erence and ?nite-volume discretizations on structured rectangular grids lead to highly e?cient numerical solvers for partial di?erential equations, yet one disadvantage of this approach is its di?culty in dealing with irregular and moving boundaries. In our previous work, we have proposed a series of fourth-order ?nite-volume methods for solving the Poisson equation, the convection-di?usion equation, and the incompressible Navier-Stokes equations. In this talk, we focus on how to augment our previous solvers to domains with irregular boundaries. First, the irregular interface are represented by cubic splines, and a Boolean algorithm computes for each control volume the open area of the ?uid, where the cell-averaged unknowns are de?ned. Second, we choose for each cut cell a set of nearby cells to discretize spatial operators; this is an open problem called poised-lattice generation in approximation theory. We solve this problem by reducing it to an exact cover problem and coupling the algorithm X [3] with techniques involving the heredity principle of quasi-determinants [2, 1]. Third, the complete decouple of spatial discretization from time integration in the GePUP formulation [4] admits a straightforward treatment of the irregular interface by replacing regular stencils with poised lattices. The resulting linear systems are e?ciently solved by geometric multigrids. Finally, we will discuss how to generalize the case of static boundaries to that of moving boundaries. References [1] I. Gelfand, S. Gelfand, V. Retakh, and R. L. Wilson. Quasideterminants. Adv. Math., 193:56–141, 2005. [2] A. Heyting. Die theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation. Math. Ann., 98:465–490, 1928. [3] D. Knuth. Dancing links. Millennial Perspectives in Computer Science, 187:159, 2000. [4] Q. Zhang. GePUP: Generic projection and unconstrained PPE for fourth-order solutions of the incompressible Navier-Stokes equations with no-slip boundary conditions. J. Sci. Comput., 67:1134–1180, 2016.

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