CMS-Flow Numerical Methods

Overview

CMS-Flow has both implicit and explicit solution schemes. The explicit solver is designed for dynamic problems with extensive wetting and drying that require small computational time-steps, while the implicit solver is intended for simulating tidal- and wave-induced circulation at tidal inlets, navigation channels, and adjacent beaches. A detailed description of the numerical formulation of the explicit solver of CMS-Flow can be found in Buttolph et al. (2006) and is not repeated here. The sections below specifically refer to the implicit solver of CMS-Flow.

The implicit solver uses the SIMPLEC (Semi-Implicit Method for Pressure Linked Equations Consistent) algorithm (van Doormal and Raithby 1984) on a non-staggered grid to handle the coupling of water level and velocity. Primary variables u-, v-velocity, and water level are stored on the same set of grid points, and fluxes at cell faces are determined using a Rhie and Chow (1983) type momentum interpolation method (Wu et al. 2011). The explicit solver uses a staggered grid with velocities at the cell faces and the water levels and water depths at the cell centers (Buttolph et al. 2006). CMS-Flow also calculates salinity, sediment transport, and morphology change. The governing equations for hydrodynamics and sediment and salinity transport have similar forms which can be written as a general transport (advection-diffusion) equation. In order to avoid redundant derivations of discretized equations, the discretization of the general transport equation is described in this chapter, and the same discretization may be applied to all of the transport equations. Then, the specific solution procedures for hydrodynamics, sediment transport, and bed change are introduced.

CMS-Flow Computational Grid

The explicit time scheme in CMS-Flow supports uniform and non-uniformly spaced Cartesian grids while the implicit version of CMS-Flow supports generic Cartesian grids which can be uniform, non-uniform, or telescoping. The telescoping locally refines the mesh by splitting a cell into subcells. The only requirement imposed by the numerical methods is that the cells must have a rectangular shape. Additional requirements are imposed by the user interface which limits the variety of types of Cartesian grids to help simplify the grid generation and avoid grid quality issues. The following requirements are applied:

1. Cells can only be subdivided into four subcells.

2. Only two neighboring cells are allowed in the same direction (i.e., north, south, east, and west).

3. Cells may have a maximum of six neighbors.

4. Refinement levels must be spaced by at least one cell apart (i.e., cells that share the same corner must be one refinement level apart).

Examples of violations of the above four requirements are shown in sequential order in Figure 1 from a to d. Limiting the number of neighboring cells to six avoids excessive cell refinement and limits the band width of the coefficient matrix for the linearlized system of equations solved by the implicit solution scheme. The last two requirements avoid having excessive cell refinement which can cause numerical instabilities. The first requirement simplifies the grid generation process but may be relaxed in future versions. The last requirement is enforced for grid quality purposes and a more smoothly varying grid resolution.

Figure 1. Examples of Invalid Cartesian computational grids

The CMS-Flow grids generated in the SMS interface can be classified as uniform, non-uniform Cartesian grids, regular telescoping, and stretched telescoping grids (Figure 2).

Figure 2. Types of Cartesian grids supported by the SMS interface and CMS-Flow.

One important aspect of incompressible flow models is the location of primary variables: velocity and pressure (water level). On a staggered grid, the pressure (water level) is located at the center of cells and the u- and v-velocities are located along the faces of cells (Harlow and Welsh 1965; Patankar 1980). On a non-staggered grid, all of the primary variables are located at the center of cells. This can lead to the spatial oscillations referred to as checkerboard oscillations and model instability if the linear interpola-tion between nodes is used to determine the fluxes at cell faces. A staggered grid can more easily eliminate these oscillations when compared to a non-staggered grid; however, a non-staggered grid involves a simpler source code and can minimize the number of coefficients that must be computed and stored during a simulation because many of the terms in the equations are equal. In particular, a staggered grid is more complicated when handling the interface between coarse and fine cells where 5- or 6-face control volumes are used. Therefore, a non-staggered (collocated) grid approach is adopted for CMS-Flow with a Rhie and Chow (1983) momentum interpolation technique used to eliminate the checkerboard oscillations. Figure 3 shows the location of primary variables on the 5- and 7-point stencils (computational molecule).

Figure 3. Computational stencils and control volumes for two types of Cartesian grids: non-uniform Cartesian (left) and telescoping grid (right).

The data structure for a grid can be approached in three ways: 1) block-structured, 2) hierarchical tree, and 3) unstructured. The block-structured approach divides the domain into multiple blocks, and each block is treated as structured. The block structured approach requires a special treatment between blocks to ensure mass and momentum balance using this approach. The hierarchical tree approach is memory intensive and requires parent-child relationships and a tree traverse to determine mesh connectivity. For the unstructured approach, all cells are numbered in a 1D sequence, and tables are used to determine the connectivity of neighboring cells. Among these three approaches, the unstructured approach is the most simple and is therefore applied in CMS-Flow. Computational cells are numbered in an unstructured manner via a 1D index array. Inactive cells (permanently dry) are not included in the 1D index array to save memory and computational time. All active computational cells are numbered sequentially. It is noted that using an unstructured data approach does not limit the model from using matrix solvers designed for structured grids. For convenience with handling boundary conditions, each boundary cell has a neighboring ghost cell outside of the computational domain.

General Transport Equation

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underbrace { \frac {\partial}{\partial t}\left(\frac{h\phi}{\beta}\right) }_{\text {Temporal Term}} + \underbrace {\frac{\partial(hV_j \phi)} {\partial x_j} }_{\text{Advection Term}} = \underbrace { \frac{\partial}{\partial x_j} \left(\Gamma h \frac{\partial \phi}{\partial x_j} \right) }_{\text{Diffusion Term}} + \underbrace{S}_{\text{Source Term}} }

(1)

where:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} = general scalar

h = total water depth [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta} = correction factor [-]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_j} = total flux velocity [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma} = diffusion coefficient for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} [Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m^2 /s} ]

S = source/sink term which includes all remaining terms.

Note that in the case of the continuity and momentum equations, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} is equal to 1 and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_i} respectively. In the case of sediment and salinity transport, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} is equal to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{sal}} , respectively. The correction factor (β) is equal to the total-load correction factor and equal to 1 for all other equations.

Spatial Discretization

A control-volume technique is used in which the governing equations are integrated over a control volume to obtain an algebraic equation that can be solved numerically. Integration of Equation (1) over a control volume yields:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_A \frac {\partial}{\partial t}\left(\frac{h_\phi}{\beta} \right) dA + \int_A \frac {\partial}{\partial x_j}\left(hV_j \phi - \Gamma h\frac{\partial \phi}{\partial x_j} \right)dA = \int_A SdA }

(2)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial}{\partial t}\left(\frac{h_P \phi_P}{\beta_P} \right) \Delta A_P + \oint_L h[(\hat{n_i}V_i)\phi - \Gamma(\hat{n_i}\bigtriangledown_i \phi)]dL = S_P \Delta A_p }

(3)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial}{\partial t}\left(\frac{h_P \phi_p}{\beta_p} \right) \Delta A_P + \sum_f h_f \left [V_f \phi_f - \Gamma_f(\bigtriangledown_\perp \phi)_f \right]\Delta l_f = S_P \Delta A_p }

(4)

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{n_i} = (\hat{n_1} , \hat{n_2})} = outward unit vector normal to cell face f [-]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_f = (\hat{n_i}V_i)_f} = outward cell face velocity [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_P} = value of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} at the cell centroid

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_P} = total water depth at the cell face centroid [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_f} = interpolated total water depth at cell face f [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\bigtriangledown_\perp \phi)_f} = outward normal gradient of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} at cell face f:Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \ (\bigtriangledown_{\perp} \phi)_f = (\hat{n_i}\bigtriangledown_i \phi)_f}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma_f} = interpolated diffusion coefficient for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta A_P} = area of cell P [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta l_f} = length of cell face f [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_P} = source/sink term.

In the above equations, the Gauss Divergence Theorem has been used to convert the area integral to a boundary integral. In addition, the area ingegrals have been evaluated assuming variable vary linearly within cells. The cell face velocity,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (V_f)} , is calculated using a momentum interpolation method similar to that of Rhie and Chow (1983) and is described in Hydrodynamics of the present Chapter.

Temporal Discretization

The general transport equation is rewritten as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial}{\partial t} \left(\frac{h\phi}{\beta} \right)= \hat{G}}

(5)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{G}} includes all the remaining terms. For stability and efficiency, a fully implicit time-stepping scheme is used in the form

where includes all the remaining terms, such as …. For stability and efficiency, a fully implicit time-stepping scheme is used in the form

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\Delta t}\left[(1 + 0.5\hat{\theta)}\frac{h^{n+1}\phi^{n+1}}{\beta^{n+1}} - (1+\hat{\theta}) \frac{h^n \phi^n}{\beta^n} + 0.5\hat{\theta}\frac{h^{n-1}\phi^{n-1}}{\beta^{n-1}} \right] = \hat{G}^{n+1} }

(6)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\theta}} is a weighting factor between 0 and 1. For Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\theta} = 0} , the scheme reduces to the first-order backward Euler scheme, and with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\theta} = 1} , the scheme reduces to the second-order backward scheme (Ferziger and Peric 1997). The superscripts indicate the time step levels, with n+1 being the next time-step and n being the previous time-step.

Cell-face Interpolation Operator

The general formula for estimating the cell-face value Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_f} is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_f = f_\perp \phi_N + (1-f_\perp)\phi_P + f_\perp(r_\parallel \bigtriangledown_\parallel \phi)_N + (1-f_\perp) (r_\parallel \bigtriangledown_\parallel \phi)_P }

(7)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_\perp} is a linear interpolation factor, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bigtriangledown_\parallel } is the gradient operator in the direction parallel to face f, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r_\parallel} is the distance from the cell center to the ghost point (O) parallel to the cell face (f) (see Figure 4).

Figure 4. Schematic showing interface interpolation: (a) coarse to fine and (b) fine to coarse.

The subscripts Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \|} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \perp} indicate the components parallel and normal to the cell face. By definition,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \| = 2|\hat{n_1}| + 1|\hat{n_2}| \text{ and }\perp = 1|\hat{n_1}| + 2|\hat{n_2}|} . Note that for neighboring cells without any refinement, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r_\|} is equal to zero and the above equation is consistent with non-refined cell faces. The linear interpolation factor is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_\perp = \frac {x_{\perp ,f} - x_{\perp,P}} {x_{\perp N} - x_{\perp ,P}} = \frac {\Delta x_{\perp ,P}} {\Delta x_{\perp , P} + \Delta x_{\perp , N}} }

(8)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_{\perp ,f}} is the coordinate of f perpendicular to the face and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta x_\perp} is the cell dimension perpendicular to the face f.

Cell-face Gradient

A linearly exact second-order approximation for the normal gradient at cell face f is calculated using the auxiliary node concept of Ferziger and Peric (1997):

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\bigtriangledown_{\perp}\phi)_f = \frac{\phi_N - \phi_P}{|\delta_{\perp}|} + \frac{(r_\|\bigtriangledown_\|\phi)_N - (r_\|\bigtriangledown_\|\phi)_P} {|\delta_{\perp}|} }

(9)

where the subscripts P and N refer to two neighboring cells, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\delta_{\perp}| = |x_{\perp,N} - x_{\perp,P}| } is the distance between cells P and N, normal to the cell face (Figure 4), and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bigtriangledown_\|} is the gradient operator in the direction parallel to face f. Ham et al. (2002) compared the auxiliary node formulation to the fully-unstructured discretization proposed by Zwart et al. (1998) for the viscous terms and found that the auxiliary node formulation is significantly more stable.

Cell-centered Gradient

The cell-centered gradient operator is calculated using the Gauss’ Divergence Theorem as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_A \bigtriangledown_i \phi dA = \sum_f \hat{n_i}\phi_f \Delta l_f}

(10)

The equation above is second order and conservative for regular and non-uniform grids.

Reconstruction, Monotonicity, and Slope Limiters

Variables are linearly reconstructed within the cells as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi = \phi_P + \overrightarrow{r}\cdot \bigtriangledown \phi_P }

(11)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_P} is the cell-average value specified at the cell centroid, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{r}} is the distance vector from the cell centroid to any location within the cell, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bigtriangledown \phi_P} is the cell-centered gradient. The reconstruction is second order and conservative in the sense that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_P \Delta A_P = \int_A \phi dA} . The linear reconstruction is used when interpolating cell-face values (Equation 7) and calculating cell-face gradients (Equation 13). If the reconstruction satisfies the local maximum principle

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle min(\phi - \phi_P,0) \leq \overrightarrow{r}\cdot \bigtriangledown \phi_P \leq max(\phi - \phi_P,0) }

(12)

then no new extrema are created within the cell, and the solution is monotonic. Figure 5 shows two examples of linear reconstruction with and without slope limiters to ensure monotonicity.

Figure 5. Schematics showing examples of (a) non-limited and (b) limited linear reconstructions.

For non-telescoping grids, the slope limiter is calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Phi(R) = \left\{\begin{align} &\frac {4R} {(R+1)^2}\quad van Leer (1979) \\ &\frac {2R} {(R^2 +1)}\quad van Albada \ et\ al.\ (1982) \\ &min \left(1, \frac{4}{R+1}, \frac {4R} {R+1} \right) \quad MUSCL (van Leer 1979) \end{align}\right. }

(13)

where R is the ratio between two consecutive slopes

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R = \frac {(\phi_{i+1}-\phi_i)(x_i - x_{i-1} ) } {(x_{i+1}-x_i)(\phi_i -\phi_{i-1})}}

(14)

Here, the second-order van Leer (1979) limiter is used because of its smoothness. Figure 6 shows a comparison of three different common limiters. The slope limiter is applied in each direction separately.

Figure 6. Comparison of three different slope limiters.

For unstructured grids the slope limiters described above are difficult to implement because of the difficulty in defining forward and backward differences. For telescoping grids the following Limited Central Difference (LCD) slope limiting procedure of Hubbard (1999) is applied:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Phi_f = \left\{\begin{align} &\frac {max(\phi_N - \phi_P , 0)} {(\overrightarrow{r}\cdot \bigtriangledown \phi)_P} &for\ (\overrightarrow{r}\cdot \bigtriangledown \phi)_P > max(\phi_N - \phi_P,0) \\ &\frac{min(\phi_N - \phi_P , 0)} {(\overrightarrow{r}\cdot \bigtriangledown \phi)_P} &for\ (\overrightarrow{r}\cdot \bigtriangledown \phi)_P < min(\phi_N - \phi_P,0) \\ &1&otherwise\\ \end{align}\right.}

(15)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{r}_{p} = \overrightarrow{x}_f - \overrightarrow{x}_p} . In the procedure outlined by Hubbard (1999), a scalar limiter is then calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Phi = min(\Phi_f)} . Here, a directional limiter is applied as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{i} = \min\limits_{\textit{f} \in \textit{f} \perp i} (\phi_f) } , which is less dissipative. Finally, the cell-centered gradient is limited as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bigtriangledown_i \phi_P = \Phi_i \bigtriangledown_i^* \phi_P}

(16)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bigtriangledown_i^* \phi_P} is the unlimited gradient.

Advection Schemes

Hybrid Scheme

The hybrid scheme is composed of a first-order upwind scheme and a second-order central difference scheme. The cell-face advective value is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_f = \left\{\begin{align} &(\phi_D + \phi_C) / 2 &for\ |P_f| > 2 \\ &\phi_c &for\ |P_f| >2 \end{align}\right. }

(17)

where the subscripts D and C indicate the downstream and upstream values, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_f = V_f|\delta_{\perp}| / \Gamma_f}  is the Peclet number at the cell face in which ,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\delta_{\perp}| = |x_{\perp,N} - x_{\perp,P}|} . In the hybrid scheme, when the Peclet number is larger than 2, the first-order upwind value is used; otherwise, the second-order central difference value is used.

Exponential Scheme

The exponential scheme interpolates the face value using an exact solution to the 1D steady advection-diffusion equation:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\phi_f - \phi_p}{\phi_N - \phi_P} = \frac{exp(P_f f_{\perp})-1}{\exp(P_f)-1} }

(18)

The exponential scheme has automatic upwinding and is stable but is usually less than second order.

Hybrid Linear/Parabolic Scheme

The Hybrid Linear/Parabolic Approximation (HLPA) scheme of Zhu (1991) may be written as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_f = \left\{ \begin{align} &\phi_c + (\phi_D - \phi_C)\hat{\phi_C}, &for\ 0 \leq \hat{\phi_C} \leq 1 \\ &\phi_C &otherwise \end{align}\right.}

(19)

where the subscripts D, C, and U indicate the first downstream and first and second upstream cells, respectively. The normalized variable Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\hat{\phi_C}} ) is determined based on the formulation of Jasak et al. (1999)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\phi_C} = \frac{\phi_C - \phi_U}{\phi_D - \phi_U} = 1 - \frac{\phi_D - \phi_C}{2(\bigtriangledown_{\perp}\phi)_C \delta_{\perp,C}}}

(20)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta_{\perp,C} = x_{\perp,D} - x_{\perp,C}} . The HLPA scheme is second order. Choi et al. (1995) found that the HLPA scheme has similar accuracy to the third-order SMARTER (Sharp and Monotonic Algorithm for Realistic Transport Efficiently Revised) and LPPA (Linear and Piecewise-Parabolic Approximation) schemes but is simpler and more efficient (Shin and Choi 1992; Choi et al. 1995).

Source/sink Term

The source/sink term is linearized as (Patankar 1980)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_P = S_P^C + S_P^P \phi_P}

(21)

The coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_P^P} is required to be non-positive for stability.

Assembly of Algebraic Equations

Assembly refers to the process of combining terms to create a linear sys-tem of algebraic equations. The algebraic equation for each cell is obtained by first combining or assembling all of the terms. Then, the continuity equation is multiplied by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_p^{n+1}} and is subtracted from the transport equation. The resulting discretized equation for cell P is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_P \phi_P^{n+1} = \sum_N a_N \phi_N^{n+1} + b }

(22)

where the subscript N refers to the neighboring cell sharing cell face, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_P} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_N} are linear coefficients for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_P^{n+1}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_N^{n+1}} . The last term, (b) contains all the remaining terms. Applying a similar equation for all of the internal cells of a grid results in a system of algebraic equations. This set of equations are referred to as the discretized governing equations.

Implicit Relaxation

The right-hand side of the algebraic system of equations (b) generally contains deferred corrections and source terms which must be computed iteratively. This iteration loop is referred to as the outer loop, and the loop within the linear iterative solver (see Section Iterative Solvers) is referred to as the inner loop. Under-relaxation is often applied to stabilize the convergence of the outer iteration loop. Under-relaxation may be applied as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi^{m+1} = a_\phi \phi^{n+1} + (1 - a_\phi)\phi^m} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_\phi} is an under-relaxation parameter Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (0 < a_\phi \leq 1)} . The superscript m indicates the previous iteration, and superscript m+1 indicates the new relaxed value. A better approach is used here to introduce the under-relaxation directly in the algebraic system of equations (Majumdar 1988; Ferziger and Peric 1997) as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {a_P}{\alpha_\phi}\phi_P^{n+1} = \sum_N a_N \phi_N^{n+1} + b + \frac {1-\alpha_\phi} {\alpha_\phi} a_P \phi_P^m }

(23)

This is referred to implicit under-relaxation and has the effect of making the coefficient matrix more diagonally dominant, thus improving convergence.

Iterative Solvers

The selection of an iterative solver is one key issue concerning the overall per-formance of the model. CMS has three iteration solvers available and they are described in more detail below: 1) GMRES variation, 2) BiCGStab, and 3) Gauss-Seidel. The default iterative solver for CMS is a variation of the GMRES (Generalized Minimum RESidual) method (Saad 1993) and is used to solve the algebraic equations. The original GMRES method (Saad and Schultz 1986) utilizes the Arnoldi process to reduce the coefficient matrix to the Hessenburg form and minimizes the norm of the residual vector over a Krylov subspace at each iterative step. The variation of the GMRES method recommended by Saad (1993) allows changes in preconditioning at every iteration step. An Incomplete LU Factorization (ILUT; Saad, 1994) is used as the preconditioner to speed-up convergence. The GMRES solver is applicable to symmetric and non-symmetric matrices and leads to the smallest residual for a fixed number of iterations. However, the memory requirements and computational costs become increasingly expensive for larger systems.

The BiCGStab (BiConjugate Gradient Stabilized) iterative solver is also a Krylov subspace solver and is applicable to symmetric and non-symmetrix matrices (Saad 1996). BiCGStab also uses ILUT as a preconditioner (Saad 1994). The BiCGStab method can be viewed as a combination of the standard Biconjugate Gradient solver where each iterative step is followed by a restarted GMRES iterative step. One advantage of the BiCGStab iterative solver is that the memory requirements are constant for each iteration and there are less computational costs when compared to the GMRES method (for less than four iterations).

The SIP (Strongly Implicit Procedure) iterative solver uses an Incomplete Lower Upper decomposition, which approximates the exact Lower Upper decomposition (Stone 1968). The method is specifically designed for algebraic systems of equations derived from partial differential equations. The implementation here is for a 5-point stencil and, therefore, only applies to nontelescoping grids.

The ICCG (Incomplete Cholesky preconditioned Conjugate Gradient) iterative solver is applicable to symmetric matrices such as the pressure correction equation (Ferziger and Peric 2002). The implementation here is also for a 5-point stencil and, therefore, can only be applied to nontelescoping grids.

The simplest iterative solver implemented in CMS is the point-implict Gauss-Seidel solver. This method may be applied in CMS with or without Succesive-Over-Relaxation to speed-up convergence (Patankar 1980). Even though the Gauss-Seidel method requires more iterations for convergence, the overall efficiency may be higher than the GMRES and BiCGStab because each itera-tion is computationally inexpensive and the code is parallelized. However, the GMRES and BiCGStab methods are more robust and perform better for large time-steps.

Convergence and Time-Stepping

During the iterative solution process, error is calculated and used to de-termine if a solution has converged, diverged, or stalled at an error below a set tolerance threshold. Anestimate of the error in solving the general algebraic equation is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r_p = \frac {1}{a_p} \left(\sum_N a_N \phi_N^{n+1} - a_P \phi_P^{n+1} + b_\phi \right)}

(24)

The $l^{2}$-norm of the residual is given by

$\|r\|_{2}={\sqrt {\sum _{cellsP}r_{P}^{2}}}$

(25)

Since this value depends on the total number of cells, the final statistic (referred to as the residual) that is used for estimating the model convergence is obtained by dividing the l²norm by the square root of the number of cells:

$R^{m}={\frac {\|r\|_{2}}{\sqrt {N_{c}}}}$

(26)

where N_c is the number of computational cells. Simply stated, R^m is referred to as the normalized residual error, and the superscript refers to the iteration number. Also, R^m is calculated for each variable that is solved at each iteration step of the solution process. Each equation has default maximum tolerances for determining if the solution has converged, diverged, or stalled. The maximum number of iterations that is imposed is set equal to M. A minimum of five iterations is required for the hydrodynamic equations, and a minimum of M/2 iterations is required for the sediment transport equations. Table 1 lists the default criteria to determine whether the iterative solution procedure has converged, requires a reduced time-step, or has diverged.

Table 1. Default criteria to determine whether the iterative solution procedure has converged, diverged, or requires a reduced time step.

Variable	Converged	Diverged	Reduce Time Step
Current velocity, m/s	If Rm < 1x10-7	If RM > 1.0x10-2 or |Ui| > 10	If RM > 1.0x10-3
Pressure- correction, m2/s2	If RM < 1x10-8	If RM > 1.0x10-3 or |p| >50	If RM>1.0x10-4
Total-load concentration, kg/m3	If RM<1x10-8	If Rm>1.0x10-3 or Ctk<0	None
Salinity, ppt	If Rm<1x10-6	If Csal < 0	None

For the implicit model, the time steps for the hydrodynamics, sediment and salinity transport are the same in order to avoid mass conservation problems and for simplicity. If any of the time step reduction criteria are met, then the time step is reduced by half and a minimum number of 3 time steps are calculated at the newly reduced time step. If the last time step converged properly, then the time step is increased. The maximum time step allowed is equal to the user-specified initial time step.

Ramp Period

For most coastal applications, the model is initialized from a “cold start”, which means that the water level and current velocities are initially set to zero. The ramp period allows the model to slowly transition from the ini-tial condition without “shocking” the system. In CMS, the ramp function is defined as

$f_{Ramp}={\frac {1}{2}}-{\frac {1}{2}}cos\left[\pi \ min(t/t_{Ramp},1)\right]$

(27)

where t is the simulation time and $t_{Ramp}$ is the ramp duration. The ramp function provides a smooth function for transitioning from the initial conditions and is plotted in Figure 7.

Figure 7. Ramp function used in CMS

The ramp function is applied to the model forcing conditions, including the wave forcing $\tau _{i}^{w}$ , surface wind $\tau _{i}^{s}$ , sediment concentration capacity $C_{tk*}$ , and significant wave height $H_{s}$ , by direct multiplication of these parameters by the ramp function at each time step during the ramp period. Boundary conditions in CMS are specified without consideration of this ramp period; therefore, the boundary conditions are also slowly transitioned from the initial condition by direct multiplication of the boundary conditions by the ramp function at each time step during the ramp period.

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