GenCade:Numerical Stability

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The GenCade model utilizes an explicit solution scheme. The main advantages of this scheme are: easy programming, simple (and sometimes the only possible) expressions of boundary conditions, and shorter computer run-time as compared to an implicit scheme (for a single time increment). A major disadvantage is, however, the stability of the solution. This means that smaller time steps are often needed, and thus a larger number, when using an explicit scheme as compared to using an implicit scheme. As a consequence, simulations using explicit schemes often require a longer total computation time. To minimize the computational effort, the longest time step that may be used for a specific calculation must be determined. Under certain idealized conditions, the CERC equation can be reduced to a simpler form to examine the dependence of the solution on the time and space steps. First, rewrite the CERC equation 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 Q = Q_O [\alpha_1 sin2 \alpha_{bs} - a_2cos(\alpha_{bs}){\frac{\partial H_b}{\partial x}}] }

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 Q_O = H^2_b C_{gb} } (cubic meters/second). A useful approximate stability criterion can be obtained by linearizing the governing equation with respect to y. The linearization is made by assuming small breaking wave and shoreline angles, which leads 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 sin2 \alpha_{bs} \approx 2 \alpha_{bs} } 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 \alpha_{bs} = \alpha_b - atan{\frac{\partial y}{\partial x}} \approx \alpha_b - {\frac{\partial y}{\partial x}} }

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 \alpha_{bs} } is the angle of breaking waves to the local shoreline orientation, 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 \alpha_{b} } is the angle of breaking waves to the x-axis, 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 {\frac{\partial y}{\partial x}} } is the local shoreline orientation. Assuming 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 {\frac{\partial q}{\partial x}} } is zero, the governing equation becomes (Kraus and Harikai 1983):

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 y}{\partial t}} = (\varepsilon_1 + \varepsilon_2) {\frac{\partial^2 y}{\partial x^2}} }

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 \varepsilon_1 = \frac{2Q_O a_1}{D_B + D_C} } 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 \varepsilon_2 = \frac{Q_O a_2 sin\alpha_b}{D_B + D_C} {\frac{\partial H_b}{\partial x}} }

In the presence of an external current not generated by breaking waves, the second diffusion coefficient in the above partial differential will change 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 \varepsilon_2 = \frac{Q_O sin\alpha_b}{D_B + D_C} (a_2 \frac{\partial H_b}{\partial x} - a_3 \frac{\bar{v}_t}{u_m}) }

As the above partial differential is a diffusion-type equation, its stability properties are well known. The numerical stability of the calculation scheme is governed 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_s = \frac{\Delta t (\varepsilon_1 + \varepsilon_2)}{(\Delta x)^2} }

where the quantity 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_s } is known as the Courant number in numerical methods; here it is called the stability parameter. The finite difference form of the above governing partial differential equation shows 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 \Delta y \approx \frac{\Delta t}{(\Delta x)^2} } which means that if 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 } is reduced by a factor of two, the time step will need to be reduced by a factor of four to maintain the same stability of the calculation scheme. If an explicit solution scheme is used to solve the diffusion equation, the following condition must be satisfied (Crank 1975):

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_s \le 0.5 }

If the 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 R_s } exceeds 0.5 at any point on the grid, the calculated shoreline will show an unphysical oscillation that will grow in time if 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_s } remains above 0.5, alternating in direction at each grid point. The quantities 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 \varepsilon_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 \varepsilon_2 } can change greatly alongshore since they depend on the local wave conditions. Assuming that the grid cell spacing is fixed by engineering requirements, a large wave height would necessitate a small 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 \Delta t } . The GenCade model will issue a warning if the stability requirement is violated at any point in the scheme. If such a warning is issued, either the time step 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 t } or the spatial resolution 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 x} } (or both) will need to be reduced. Thus, it is necessary to increase 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 t } and/or decrease 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 } .

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