Qinghai Zhang
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MATHEMATICS COLLOQUIUM
Speaker: Qinghai Zhang Abstract. An approximate projection method is proposed for numerically solving the incompressible Navier-Stokes (INS) equations on adaptively refined no-slip domains with fourth-order accuracy both in time and in space. Via an estimate of the Laplace-Leray commutator, the velocity is formulated as the only evolutionary variable; the divergence-free constraint is replaced by a pressure Poisson equation so that the dissipation of velocity divergence is governed by the heat equation. Within each time step, the solution is advanced by solving a sequence of linear systems via geometric multigrid methods. To achieve certain accuracy requirements, the proposed method running on a personal desktop is faster than some classical second-order methods running on the fastest supercomputer in this world! As the first step to generalize the INS solver to domains with moving boundaries, I present a new interface tracking method, called the polygonal area mapping method (PAM), which explicitly represents material areas as piecewise polygons, traces polygon boundaries along pathlines, and calculates new material areas via Boolean operations on polygons. Recently, PAM is further improved to the fourth-, sixth-, and eighth-order accuracy via an analytic framework of mapping and adjusting regular semi-algebraic sets. Compared with other interface tracking methods, PAM has a number of notable advantages such as (i) direct applicability to both structured and unstructured grids, (ii) high accuracy and efficiency, and (iii) easy extension to an arbitrary number of materials. For practical applications, the physics of different types of interface must be incorporated into the numerical scheme; this is illustrated by a variety of geophysical and biological multiphase flows such as air-water free-surface flows, crustacean swimming, and blood clotting. Overall, the cost of the proposed computational framework within each time step is linear in terms of the number of unknowns. The combination of high-order convergence rates, parallel computing, and adaptive mesh refinement pushes the resolving power of the proposed framework to the limit of a given hardware facility. Future plans for further expanding its application range will also be discussed. |