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:

$Q=Q_O[{\alpha }_{1}sin2{\alpha }_{bs}-a_{2}cos({\alpha }_{bs})\frac{\partial H_b}{\partial x}]$

where $Q_O=H_b^2{C}_{gb}$ (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:

$sin2{\alpha }_{bs}\approx 2{\alpha }_{bs}$ and ${\alpha }_{bs}={\alpha }_b-atan\frac{\partial y}{\partial x}\approx {\alpha }_b-\frac{\partial y}{\partial x}$

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

$\frac{\partial y}{\partial t}=({\epsilon }_1+{\epsilon }_2)\frac{{\partial }^{2}y}{\partial x^2}$

where ${\epsilon }_1=\frac{2Q_O{a}_1}{D_B+D_C}$ and ${\epsilon }_2=\frac{Q_O{a}_{2}sin{\alpha }_b}{D_B+D_C}\frac{\partial H_b}{\partial x}$