Roller Numerical Methods

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

$\frac{\partial (S_{sr}{c}_j)}{\partial x}{|}_{i,j}=\{\begin{array}{rl}& \frac{(S_{sr}{c}_j{)}_{i,j}-(S_{sr}{c}_j{)}_{i,j-1}}{\delta x_{i,j-1}},\text{for }{c}_{i,j}>0\\ & \frac{(S_{sr}{c}_j{)}_{i,j+1}-(S_{sr}{c}_j{)}_{i,j}}{\delta x_{i,j}},\text{for }{c}_{i,j}<0\end{array}$

(1)

where $S_{sr}=2E_{sr}$ and i and j indicate the position along either the rows or columns, and $\delta x_{i,j}$ is the cell-center distance between adjacent cells in the j^th direction and at position i. The second-order upwind scheme is given by

$\frac{\partial (S_{sr}{c}_j)}{\partial x_j}{|}_{i,j}=\{\begin{array}{rl}& \frac{3(S_{sr}{c}_j{)}_{i,j}-4(S_{sr}{c}_j{)}_{i,j-1}+(S_{sr}{c}_j{)}_{i,j-2}}{\delta x_{i,j}+\delta x_{i,j-1}},\text{for }{c}_{i,j}>0\\ & \frac{-3(S_{sr}{c}_j{)}_{i,j}+4(S_{sr}{c}_j{)}_{i,j+1}-(S_{sr}{c}_j{)}_{i,j+2}}{\delta x_{i,j}+\delta x_{i,j+1}},\text{for }{c}_{i,j}<0\end{array}$

(2)

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

$(S_{sr}{)}^{n+1}=(S_{sr}{)}^n+\mathrm{\Delta }{t}_{sr}{(-D_r+f_e{D}_{br}-\frac{\partial (S_{sr}{c}_j)}{\partial x})}^n$

(3)

where ${\mathrm{\Delta }}_{sr}$ is the surface roller time step and is determined as $\mathrm{\Delta }{t}_{sr}=\text{0.5 max}(\mathrm{\Delta }{x}_j/c)\text{ where }\mathrm{\Delta }{x}_j$ is the cell size in the $j^{th}$ 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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