Roller Numerical Methods

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The surface roller transport equation is solved in CMS-Wave 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 first-order or second-order upwind finite difference scheme. The first order upwind scheme is given by



where and i and j indicate the position along either the rows or columns, and is the cell-center distance between adjacent cells in the jth direction and at position i. The second-order upwind scheme is given by



The surface roller calculation is achieved by setting the initial roller energy and time-stepping until the steady-state solution is reached. For simplicity, an explicit Euler scheme is used as follows



where is the surface roller time step and is determined as is the cell size in the direction. The steady-state solution is typically reached after ~40-80 time steps and takes about 1-2 seconds to run on a desktop personal computer.

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