High Order Numerical Methods for PDEs on Complex Domains

The goal of this project is to develop efficient and robust high order accuracy numerical methods for solving hyperbolic conservation laws, Hamilton-Jacobi equations and stiff advection-reaction-diffusion equations defined on complex domains. The methods include Weighted ENO methods, discontinuous Galerkin methods, fast sweeping methods and high order time-stepping implicit integration factor methods.

Collaborators

  • Dr. Qing Nie (Mathematics and Biomedical Engineering, UC Irvine)
  • Dr. Chi-Wang Shu (Applied Mathematics, Brown University)
  • Dr. Yongtao Zhang (ACMS, Notre Dame)
  • Dr. Hongkai Zhao (Mathematics, UC Irvine)


1) From the paper “S. Chen and Y.-T. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods, Journal of Computational Physics, v230, (2011), pp. 4336-4352.”

high_order_pde1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


2) From the paper “Y.-T. Zhang, S. Chen, F. Li, H. Zhao and C.-W. Shu, Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations, SIAM Journal on Scientific Computing, v33, no. 4, (2011), pp. 1873-1896.”

Shape-from-shading problem in computer vision:

high_order_pde2


3) From the paper “J. Zhu, Y.-T. Zhang, M. S. Alber and S. A. Newman, Bare bones pattern formation: a core regulatory network in varying geometries reproduces major features of vertebrate limb development and evolution, PLoS ONE, v5(5): e10892, (2010).”

Simulation of fossil limb skeletons:
 

high_order_pde3

 

(4) From the paper “Y.-T. Zhang, C.-W. Shu and Y. Zhou, Effects of Shock Waves on Rayleigh-Taylor Instability, Physics of Plasmas, v13, (2006), article number 062705.”

Simulation of interactions of shock waves with Rayleigh-Taylor flow, by 9th order WENO schemes:

high_order_pde4