Moving Mesh Method for Singular Problems

Numerical solution of many problems requires small grid size over a portion of the physical domain to resolve large solution variations. Using a uniform mesh for these problems is formidable when the system involves two or more spatial dimensions. In the adaptive mesh method, the total number of grid points is fixed while their locations change according to the evolution of the physical system - the grid points move towards the regions where large solution variations develop. We developed an efficient grid redistribution method in multiple dimensions and applied the method to study the self-focusing phenomenon of the non-linear Schrodinger equation.

• Numerical simulation of self-focusing of ultrafast laser pulse, G. Fibich, W. Ren and X. P. Wang, Phys. Rev. E 67, 056603 (2003) 
• A new adaptive grid method based on iterative grid redistribution, W. Ren and X. P. Wang, Methods and Application of Analysis, 8, 515 (2001)
• An iterative grid redistribution method for singular problems in multiple dimensions, W. Ren and X. P. Wang, J. Comput. Phys. 159, 246 (2000)