we develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. this model describes the cell polarity and movement using a reaction-diffusion system coupled with a sharp-interface model. 

we employ an efficient finite difference method for the reaction-diffusion equations with no-flux boundary conditions. this results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. to track the long-time dynamics, we employ techniques of the moving computational window to keep the efficiency. our level-set simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features. 

our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important. this is a joint work with li-tien cheng and bo li.