Research projects in CCMA

　
• Software projects for the theory of partitions, (George Andrews)

  • Using computer algebra systems available, the MacMahon's partitions and also the generalized Engel algorithm will be investigated and applied for the theory of partitions.

• High order numerical methods for hyperbolic conservation laws by vanishing viscosity, (Alberto Bressan, Wen Shen, Jinchao Xu and Zhengfu Xu)

  • High order numerical methods will be developed for hyperbolic conservation laws, using the latest theoretical results of vanishing viscosity together with multigrid and extrapolation techniques.

• Modeling and simulations of multiscale problems in complex fluids and biology, (Qiang Du, Chun Liu and Andrew Belmonte)

  • Scientific computation and visualization play critical roles in much of our on-going research collaboration, ranging from investigating the geometric deformation and topological transformation of elastic cellular membranes, studying complex viscoelastic fluids, to understanding of defects in liquid crystals and quantized vortices in superconductors and Bose-Einstein condensate.

• Computation with a nonlinear dynamic programming problem, (Jenny Li)

  • A high dimensional nonlinear optimization problem that involves the Bellman equation will be numerically solved for the analysis of the main sources of financial sector vulnerability which include credit risk, market risk and liquidity risk.

• Modeling mechanical behavior of solids, (Xiantao Li)

  • A multiscale method will be developed for crystalline solids, including a domain decomposition technique that divides the computational domain into continuum regions where nonlinear elasticity is applied and atomistic regions where molecular statics/dynamics is used to reflect microscale configuration. The atomistic and continuum models are coupled through a transparent boundary condition at the interface.

• Algebraic multigrid for fully discrete physical models, (Xiantao Li, Alexei Novikov, Jinchao Xu and Ludmil Zikatanov)

  • Efficient adaptive algebraic multigrid (AMG) methods will be developed and applied for material studies based on atomistic models, such as molecular mechanics and asymptotic models describing the transport of plasma proteins in blood.

• Improved Subgrid-Scale Modeling of Turbulent flows, (Alexei Novikov)

  • Analytical and numerical multiscale results will be implemented to accurately model the effect of small scales on large-scale flows.

• Adaptive and multigrid methods for simulations of fuel cells, (Pengtao Sun, Chao-Yang Wang, Jinchao Xu and Guangri Xue)

  • Advanced numerical techniques will be developed for a set of fuel cell model partial differential equations that feature anisotropy, degeneracy, discontinuity, nonlinearity and coupling between different physical components.

• Dynamics of Population Models, (H. Weiss)

  • This research incorporates two large research projects in population dynamics. Both of these projects require heavy use of high-end powerful computing resources, and the use of an SMP computer would greatly aid this research.

• Black Hole Simulation, (Carlos F. Sopuerta, Pengtao Sun, Pablo Laguna and Jinchao Xu)

• Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modeling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.

• Two Phase Flows, (Andrew Belmonte, Chun Liu, Pengtao Sun and Jinchao Xu)
• The research on this project is supported in part by the National Science Foundation: Award No. DMS-0074299, DMS-0209497, and SCREMS award No. DMS-0215392


• Mesh Adaptation, (Long Chen, Pengtao Sun and Jinchao Xu)
  • The research on this project is supported in part by the National Science Foundation: Award No. DMS-0074299, DMS-0209497, and SCREMS Award No. DMS-0215392
  
  
• Theory and application of subspace correction methods, (Jinchao Xu, Ludmil Zikatanov
)
• The research on this project is supported by the National Science Foundation: Award No. DMS-0074299 and DMS-0209497