Roller Numerical Methods
The surface roller transport equation is solved in CMSWave using a finite difference method. The source terms are calculated at the grid cell centers. The advective or transport term is approximated using either the firstorder or secondorder upwind finite difference scheme. The first order upwind scheme is given by

(1) 
where and i and j indicate the position along either the rows or columns, and is the cellcenter distance between adjacent cells in the j^{th} direction and at position i. The secondorder upwind scheme is given by

(2) 
The surface roller calculation is achieved by setting the initial roller energy and timestepping until the steadystate solution is reached. For simplicity, an explicit Euler scheme is used as follows

(3) 
where is the surface roller time step and is determined as is the cell size in the direction. The steadystate solution is typically reached after ~4080 time steps and takes about 12 seconds to run on a desktop personal computer.