Marker method for the solution of nonlinear diffusion equations

The marker method for the solution of nonlinear diffusion equations is described. The method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some detail. Numerical experiments show that the method is accurate in determining the long time behavior of nonlinear diffusion equations. The marker method can be applied to an ensemble of nonlinear dispersive partial differential equations.