Professor Jian-Guo Liu (Duke University) is a Nelder Fellow in the Department until end of June. Jian-Guo Liu is a professor at the Department of Mathematics and the Department of Physics at Duke university, USA. He received his PhD in Mathematics at UCLA in 1990. He works on applied analysis for nonlinear partial differential equations arising from fluid dynamics and kinetic theory. He develops and analyzes numerical methods for fluid dynamics and complex fluids. He also works on emergent behavior and self-organization for some complex systems from biological and social sciences. He has published more than 100 journal papers and he is a managing editor for the Journal of Hyperbolic Differential Equations and a managing editor for the Numerical Method and Analysis. Jian-Guo Liu's web page can be found here
The schedule and abstract of the course is as follows: Lectures 1 & 2: Tuesday June 10, 4:00-6:00 PM room Huxley 140 Lectures 3 & 4: Thursday June 12, 4:00-6:00 PM Clore lecture theatre Lecture 1. Projection methods for viscous incompressible flows. How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier-Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure. Reference: J.-G. Liu, J. Liu and R. Pego, Stable and accurate pressure approximation for unsteady incompressible viscous flow J. Comput. Phys. 229 (2010) 3428-3453 Lecture 2. Boundary Layer Separation and the Infinite Reynolds Number Limit of the Flow Past a Cylinder and a Sphere. We present an essentially compact 4th order scheme for axisymmetric flows past a spherical obstacle. The main ingredients of the scheme include (1) the generalized vorticity formulation that validates the divergence free constraint automatically and essentially decouples the equations, (2) proper formulation and implementation of the pole condition that guarantees the stability of the scheme under time evolution, and (3) a novel change of variable that enables a fast Poisson solver for the stream function via FFT and realization of the exact far field boundary condition. High Reynolds number simulations are presented, demonstrating that the scheme is capable of resolving fine details of the complex wake structures. Lectures 3 & 4. An analysis of merging-splitting group dynamics by Bernstein function theory We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no H-theorem, we are able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers (derived by Lambert in 1758) that count all ternary trees with n nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences as those Pick functions analytic and nonnegative on (-infinity,1). This is joint work with Pierre Degond and Bob Pego. |