题目：Central Schemes on Overlapping Cells

报告人：Yingjie Liu

单位：（School of Mathematics, Georgia Institute of Technology)

时     间： 2006年6月19日16:30~17:30

地     点： 北京大学理科一号楼1418M

Tea  Time:  2006年6月19日16:15-16:30

地     点： 北京大学理科一号楼1384E

主办单位：  北京大学科学与工程计算中心

            北京计算数学学会

Abstract:
The central scheme of Nessyahu and Tadmor (J. Comput. Phys, 87(1990))has the benefit of not having to deal with the solution within the Riemann fan for solving hyperbolic conservation laws and related equations. But the staggered averaging causes large dissipation when the time step size is small comparing to the mesh size. The recent work of Kurganov and Tadmor (J. Comput. Phys, 160(2000)) overcomes the problem by  use of a variable control volume and obtains a semi-discrete non-staggered central scheme.

Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the $O(1/\Delta t)$ dependence of the dissipation. Semi-discrete form of the central scheme can also be obtained. This technique is essentially independent of the reconstruction and the shape of the mesh, thus could also be useful for Voronoi mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions.

Generally more compact reconstruction can be achieved. We demonstrate through numerical examples that combining two classes of the overlapping cells in the reconstruction can achieve higher resolution.

Overlapping cells create self similarity in the grid and enable the development of central type discontinuous Galerkin methods, following the series works of Cockburn and  Shu (Math. Comp. 52(1989)).

If time allows, I will also discuss some recent applications of the back and forth error compensation and correction (BFECC) algorithm for Navier Stokes fluid simulations and the level set interface computation.

题目：A Lagrangian--Eulerian Approach For The Numerical Simulation Of Free-Surface Flow

报告人：Jie Li

单位：  (Department of Engineering,  University of Cambridge)

时     间： 2006年6月20日16:30~17:30

地     点： 北京大学理科一号楼1418M

Tea  Time:  2006年6月20日16:15-16:30

地     点： 北京大学理科一号楼1384E

主办单位： 北京大学科学与工程计算中心

           北京计算数学学会

Abstract:
A new numerical method specially adapted to the free-surface flow of viscoelastic material is proposed. It is based on a Lagrangian discretisation of the material and objective derivatives, which accounts well for the hyperbolic nature of these terms and goes well with a Lagrangian tracking of a time-evolving domain.

Through the Arbitrary Lagrangian-Eulerian formulation, the method can also be applied efficiently to solid-boundary problems, and is tested on the benchmark problem of the drag on a cylinder in a channel. The collapse of a column of Oldroyd-B fluid is then considered: under the action of surface tension, the column undergoes large deformation leading to the“beads-on-string”structure. Asymptotic results on the evolution of this structure are recovered in numerical simulations, and further features of this flow are exhibited.