I-Liang Chern

Front tracking for gas dynamics

Front tracking is an adaptive computational method in which a lower dimensional moving grid is fitted to and follows the dynamical evolution of distinguished waves in a fluid flow. The method takes advantage of known analytic solutions, derived from the Rankine-Hugoniot relations, for idealized discontinuities. In this paper the method is applied to the Euler equations describing compressible gas dynamics. The main thrust here is validation of the front tracking method: we present results on a series of test problems for which comparison answers can be obtained by independent methods.

Interest Rate Modeling

As pointed out in Sect. 2.3, when the short-term interest rate is considered as a random variable, there is an unknown function λ(r, t), called the market price of risk, in the governing equation.

Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates

In this paper, we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in Bose-Einstein condensates (BECs). We begin with a review on the gradient flow with discrete normalization (GFDN) for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the GFDN. For discretizing the gradient flow, we use sine-pseudospectral method for spatial derivatives and either backward Euler scheme (BESP) or backward/forward Euler schemes for linear/nonlinear terms (BFSP) for temporal derivatives. Both BESP and BFSP are spectral order accurate for computing the ground and first excited states in BEC. Of course, they have their own advantages: (i) for linear case, BESP is energy diminishing for any time step size where BFSP is energy diminishing under a constraint on the time step size; (ii) at every time step, the linear system in BFSP can be solved directly via fast sine transform (FST) and thus it is extremely efficient, and in BESP it needs to be solved iteratively via FST by introducing a stabilization term and thus it could be efficient too. Comparisons between BESP and BFSP as well as other existing numerical methods are reported in terms of accuracy and total computational time. Our numerical results show that both BESP and BFSP are much more accurate and efficient than those existing numerical methods in the literature. Finally our new numerical methods are applied to compute the ground and first excited states in BEC in one dimension (1D), 2D and 3D with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution.

Long-time effect of relaxation for hyperbolic conservation laws

The hyperbolic conservation laws with relaxation appear in many physical systems such as nonequilibrium gas dynamics, flood flow with friction, viscoelasticity, magnetohydrodynamics, etc. This article studies the long-time effect of relaxation when the initial data is a perturbation of an equilibrium constant state. It is shown that in this case the long-time effect of relaxation is equivalent to a viscous effect, or in other words, the Chapman-Enskog expansion is valid. It is also shown that the corresponding solution tends to a diffusion wave time asymptotically. This diffusion wave carries an invariant mass. The convergence rate to this diffusion wave in theLp-sense for 1≦p≦∞ is also obtained and this rate is optimal.

Acceleration Methods for Total Variation-Based Image Denoising

In this paper, we apply a fixed point method to solve the total variation-based image denoising problem. An algebraic multigrid method is used to solve the corresponding linear equations. Krylov subspace acceleration is adopted to improve convergence in the fixed point iteration. A good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid. Numerical experiments demonstrate that this method is efficient and robust even for images with large noise-to-signal ratios.

New Formulations for Interface Problems in Polar Coordinates

In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a nonsmooth or discontinuous solution into a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution. In conjunction with the fast immersed interface method, a fast solver has been developed for the interface problems with a piecewise constant but a discontinuous coefficient using the new formulation in a polar coordinate system.

Enhancing image watermarking methods with/without reference images by optimization on second-order statistics

The watermarking method has emerged as an important tool for content tracing, authentication, and data hiding in multimedia applications. We propose a watermarking strategy in which the watermark of a host is selected from the robust features of the estimated forged images of the host. The forged images are obtained from Monte Carlo simulations of potential pirate attacks on the host image. The solution of applying an optimization technique to the second-order statistics of the features of the forged images gives two orthogonal spaces. One of them characterizes most of the variations in the modifications of the host. Our watermark is embedded in the other space that most potential pirate attacks do not touch. Thus, the embedded watermark is robust. Our watermarking method uses the same framework for watermark detection with a reference and blind detection. We demonstrate the performance of our method under various levels of attacks.

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the self-similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves isO(t−3/4+1/2p+σ) inLp, 1≦p≦∞, with σ>0 being arbitrarily small.

Vorticity generation and evolution in shock-accelerated density-stratified interfaces

The results of direct numerical simulations of inviscid planar shock‐accelerated density‐stratified interfaces in two dimensions are presented and compared with shock tube experiments of Haas [(private communication, 1988)] and Sturtevant [in Shock Tubes and Waves, edited by H. Gronig (VCH, Berlin, 1987), p. 89] . Heavy‐to‐light (‘‘slow/fast or s/f) and light‐to‐heavy (‘‘fast/slow,’’ or f/s) gas interfaces are examined and early‐time impulsive vorticity deposition and the evolution of coherent vortex structures are emphasized and quantified. The present second‐order Godunov scheme yields excellent agreement with shock‐polar analyses at early time. A more physical vortex interpretation explains the commonly used (i.e., linear paradigm) designations of ‘‘unstable’’ and ‘‘stable’’ for the f/s and s/f interfaces, respectively. The later time events are Rayleigh–Taylor like and can be described in terms of the evolution of a vortex layer (large‐scale translation and rotation): asymmetric tip vortex ‘‘roll‐up’’...

Exploring ground states and excited states of spin-1 Bose-Einstein condensates by continuation methods

A pseudo-arclength continuation method (PACM) is employed to compute the ground state and excited state solutions of spin-1 Bose-Einstein condensates (BEC). The BEC is governed by the time-independent coupled Gross-Pitaevskii equations (GPE) under the conservations of the mass and magnetization. The coupling constants that characterize the spin-independent and spin-exchange interactions are chosen as the continuation parameters. The continuation curve starts from a ground state or an excited state with very small coupling parameters. The proposed numerical schemes allow us to investigate the effect of the coupling constants and study the bifurcation diagrams of the time-independent coupled GPE. Numerical results on the wave functions and their corresponding energies of spin-1 BEC with repulsive/attractive and ferromagnetic/antiferromagnetic interactions are presented. Furthermore, we reveal that the component separation and population transfer between the different hyperfine states can only occur in excited states due to the spin-exchange interactions.