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

$\frac{\partial (S_{s r} c_{j})}{\partial x} |_{i, j} = {\begin{aligned} \frac{(S_{s r} c_{j})_{i, j} - (S_{s r} c_{j})_{i, j - 1}}{δ x_{i, j - 1}}, for c_{i, j} > 0 \\ \frac{(S_{s r} c_{j})_{i, j + 1} - (S_{s r} c_{j})_{i, j}}{δ x_{i, j}}, for c_{i, j} < 0 \end{aligned}$

(1)

where $S_{s r} = 2 E_{s r}$ and i and j indicate the position along either the rows or columns, and $δ x_{i, j}$ 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

$\frac{\partial (S_{s r} c_{j})}{\partial x_{j}} |_{i, j} = {\begin{aligned} \frac{3 (S_{s r} c_{j})_{i, j} - 4 (S_{s r} c_{j})_{i, j - 1} + (S_{s r} c_{j})_{i, j - 2}}{δ x_{i, j} + δ x_{i, j - 1}}, for c_{i, j} > 0 \\ \frac{- 3 (S_{s r} c_{j})_{i, j} + 4 (S_{s r} c_{j})_{i, j + 1} - (S_{s r} c_{j})_{i, j + 2}}{δ x_{i, j} + δ x_{i, j + 1}}, for c_{i, j} < 0 \end{aligned}$

(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

$(S_{s r})^{n + 1} = (S_{s r})^{n} + Δ t_{s r} {(- D_{r} + f_{e} D_{b r} - \frac{\partial (S_{s r} c_{j})}{\partial x})}^{n}$

(3)

where $Δ_{s r}$ is the surface roller time step and is determined as $Δ t_{s r} = 0.5 max (Δ x_{j} / c) where Δ x_{j}$ is the cell size in the $j^{t h}$ 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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Roller Numerical Methods

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