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- Tao Zhou, Zoltán Kuscsik, Jian-Guo Liu, Matús Medo, Joseph Rushton Wakeling, Yi-Cheng Zhang
- Proceedings of the National Academy of Sciences…
- 2010

Recommender systems use data on past user preferences to predict possible future likes and interests. A key challenge is that while the most useful individual recommendations are to be found among diverse niche objects, the most reliably accurate results are obtained by methods that recommend objects based on user or object similarity. In this paper we… (More)

- Weinan E, Jian-Guo Liu
- 2000

This is the first of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of these papers is to provide a thorough understanding of the numerical phenomena involved in the projection methods, particularly when boundaries are present, and point to ways of designing more efficient, robust and… (More)

- Cheng Wang, Jian-Guo Liu
- Math. Comput.
- 2000

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable φ, u = a + ∇φ, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward… (More)

- Jian-Guo Liu, Wei-Cheng Wang
- SIAM J. Numerical Analysis
- 2006

We give an error estimate for the Energy and Helicity Preserving Scheme (EHPS) in second order finite difference setting on axisymmetric incompressible flows with swirling velocity. This is accomplished by a weighted energy estimate, along with careful and nonstandard local truncation error analysis near the geometric singularity and a far field decay… (More)

- WEINAN E, JIAN-GUO LIU
- 1996

context of finite difference schemes in vorticity formulation has a long history, going back at least to the 1930s when This paper discusses three basic issues related to the design of finite difference schemes for unsteady viscous incompressible Thom's formula (see (2.4)) was derived [20]. Thom's for-flows using vorticity formulations: the boundary… (More)

- JIAN-GUO LIU, ZHOUPING XIN
- 2006

We prove the convergence of vortex blob methods to classical weak solutions for the two-dimensional incompressible Euler equations with initial data satisfying the conditions that the vor-ticity is a finite Radon measure of distinguished sign and the kinetic energy is locally bounded. This includes the important example of vortex sheets. The result is valid… (More)

For strong solutions of the incompressible Navier-Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time-stepping for pressure. These schemes require no solution of stationary… (More)

- Jian-Guo Liu, Luc Mieussens
- SIAM J. Numerical Analysis
- 2010

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a… (More)

- Jeffrey Haack, Shi Jin, Jian-Guo Liu
- 2010

The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time… (More)

- Weinan E, Jian-Guo Liu
- 2000

SUMMARY A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and… (More)