Long Lee

Contact information:

Academic Positions

• 2005 - present, Assistant Professor, University of Wyoming, Laramie
• 2002 - 2005, Post-doctoral Fellow, University of North Carolina, Chapel Hill

Education

• Ph.D. Applied Mathematics, University of Washington, Seattle, 2002

Teaching

• Spring 2006, Numerical analysis Math 4340
• Spring 2006, Calculus I, section 02 Math 2200
• Fall 2005, Computational methods III Math 5345
• Spring 2005, Introduction to Differential Equations Math 83
• Fall 2004, Mathematical Methods for the Physical Sciences I Math 128
• Spring 2004, Linear Algebra for Applications Math 147

Research

Fluid dynamics at interfaces:

One of my research interests is numerical methods and numerical analysis for fluid dynamics at interfaces, especially for problems with moving immersed boundaries. Professor Randall J. LeVeque and I have developed a new fractional step method for 2-D incompressible flows based on high-resolution conservative algorithms - CLAWPACK. Using this high-resolution projection method, we are able to extend the Immersed Interface Method (IIM) to the incompressible Navier-Stokes equations with moving immersed boundaries.

Nonlinear waves:

I have done some research in developing and analyzing integrable and integral algorithms for solving a nonlinear shallow-water wave equation. The integrable algorithm results in a particle method. We establish the existence of the system of ordinary differential equations obtained from the discretization of the particle system, followed by a proof of convergence for the algorithm. A fast lattice-sum method is proposed to evaluate the integrals arising from the particle method, which makes the method robust and more efficient than the traditional pseudospectral methods. The integral algorithm is efficient and is easy to implement for higher order schemes using local corrections. This work is joint with Roberto Camassa and Jingfang Huang.

Multiphase fluids and bio-fluid dynamics:

The primary ongoing project of mine is modeling the mucus transport in human lungs. This project leads to two directions. One is characterization of the effective mucus transport in the lung. The other is the simulatations for the muco-ciliary interaction. The experiments show that mucus clearance can be achieved by two phase gas-liquid flow mechanism in patients with excessive bronchial secretions with biased tidal breathing favoring the expiatory flow. We derive an evolution equation for the interfacial waves between the mucus layer and the airway, and give a two-phase gas-liquid model for the mucus transport for the experiments. Analysis and numerical simulations are under investigation for this model. The muco-ciliary interaction is under investigation using the immersed interface method.

Publications

• (With R. J. LeVeque) An immersed interface method for the incompressible Navier-Stokes equations, vol 25 no. 3 pp 832-856 SIAM Sci. Comp.
• ( With R. Camassa and J. Huang) On a completely integrable numerical scheme for a nonlinear shallow-water wave equation J. of Nonlinear Math Phys., 12, Supplement 1, pp 146-162, 2005.
• ( With R. Camassa and J. Huang) Integral and integrable algorithms for a nonlinear shallow-water wave equation, accepted, to appear in JCP
• (with R. Camassa) Thin film dynamics in a liquid lined circular pipe To appear, Book Chapter, in honor of Professor Ted Wu.

Simulation gallery:

Periodic Particle

• kappa = 0, period = 80, initial = jacobicn gif animation

• kappa = 0, period = 2 pi, initial = cos (x) + 2 gif animation

  Shallow water double gyre model

• Double Gyre over 10 years (height field) gif animation

• The passive tracer without diffusion gif animation

• The passive tracer with diffusion (diffussivity is the same as the shallow water model) gif animation

• The initial condition (double gyres height field) jpeg image

• The initial condition of the passive tracer jpeg image

• The advection of the passive tracer jpeg image

• The advection-diffusion of the passive tracer jpeg image

  Nonlinear wave propagation:

  Two different boundary conditions

  • exact u_x:(exact BC)
  • phony boundary condition (u_x=(exact u_x)*exp(-t^2)) (phony BC)

  Lax-Wendroff scheme for the quarter-plane problem (1-D linear advection):

  • boundary condition is specified. (correct BC)
  • incorrect boundary condition (computed from u^{n+1}(1) and u^n(0)) (incorrect BC)

  Initial-boundary value problem (integral equation method):

  • traveling wave solution (I.C. centered at x=0), kappa=1, movie
  • traveling wave solution (I.C. at the left plane entirely), kappa=1, movie

  Initial-boundary value problem (integral equation method):
  boundary condition at x=0: u(0)=0

  • full domain movie1
  • blow up movie2

  ---

  Moving interface simulations:

  
  Movie 1: Pressurized balloon in driven cavity. Re=5.
  
           mpeg version
  
  
  Movie 2: Contraction of an elastic line.
  
  Movie 3: Contraction of a massless elastic membrane.
  
  Movie 3.1: Contraction of an elastic membrane with total mass 4.
  
  Movie 3.2: Contraction of an elastic membrane with total mass 32.
  
           damping plot
  
  Movie 4: Rayleigh-Taylor instability. Density ratio=1/3. Ra=1000.
  
  Movie 5: Rayleigh-Benard  convection. Axis ratio, x:y=4:1.
  
  Movie 6: Droplet splash 
  

  Feel free to send me email at llee@uwyo.edu