Discrete Scheme

The Local Inertial Approximation to the Shallow Water Equations (SWE-LIA) utilizes a simplified momentum equation which ignores the advection, diffusion, and Coliolis terms. The semi-discrete form of the second fractional step may be written as

1) \displaystyle \frac{u_{N,f}- u^n_{N,f}}{\Delta t} = -g\frac{\partial z_{s,f}^{n + \theta}}{\partial N} - c_{f,f} u_{N,f}^{n+1} + \frac{\tau _{s,N}}{\rho h_f^n}

Barotropic Pressure Gradient

In the LIA-SWE model, the barotropic pressure gradient term is treated similar the ELM-SWE model. The term is computed at computational faces utilizing the two-point stencil described above and treats the water levels semi-implicitly. The term is may be written as

2) g \nabla z_s \cdot \textbf{n}_f = g \frac{\partial z_s^{n+\theta} }{\partial N}

Solution Procedure

The momentum equation above can be rearranged to obtain an expression for the velocity at n+1 as

3) \displaystyle u_{N,f}^{n+1} = - \frac{\Delta t \theta g} {1 + \Delta t c_{f,f} } \frac{\partial z_{s,f}^{n+1}} {\partial N} + F_f

where

F_f = \frac{ B_f^n} {1 + \Delta t c_{f,f}^{n+\theta} }
B_f^n = u_{N,f}^n - (1-\theta) \Delta t g\frac{\partial z_s^n } {\partial N} + \Delta t \frac{ \tau_{s,N} } {\rho h_f^n}

To obtain a discrete implicit equation for the water volume, the above equation is inserted into the discrete continuity equation to obtain

4) \displaystyle \Omega_i^{n+1} + \sum_{j \in \text C (i)} a_{i,j} z_{s,j}^{n+1} = b_i

where

a_{i,j} = - \frac{ \Delta t^2 \theta^2 g A_k } { \left ( 1+ \Delta t c_{f,f} \right ) \Delta x_N \ }, j \in \text{N}(i)
a_{i,i} = - \sum_{j \in \text{N} (i)} a_{i,j}
b_i = \Omega_i^n + \Delta t Q_i + \frac{1 - \theta}{\theta} \sum_{j \in \text{C}(i)} a_{i,j} z_{s,j}^n + \Delta t \sum_{k \in \text{K}(i)} \left [ \theta F_k + (1-\theta) u_{N,k}^n \right ] A_k

The system of equations is solved using the same Newton-like iterations used for the DWE solver.

Discrete Boundary Conditions

The boundary conditions are discretized in manner as in the ELM-SWE solver as:

  • Water surface elevation: The water surface elevation boundary condition is directly implemented as . The internal cell is then discretized as described above and the terms containing the boundary water surfaces are placed on the right-hand-side of the system of equations.
  • Normal Depth: The friction slope, , is specified and utilized to compute a flow at each computation face as . Boundary face flows are included in the internal cells as a source term on the right-hand-side of the system of equations.
  • Flow: The flow boundary condition is specified at each computational face based on the local conveyance. Boundary face flows are included in the internal cells as a source term on the right-hand-side of the system of equations.

Solution Algorithm.

The solution algorithm of the SWE-EM solver proceeds exactly as described in the SWE-ELM solver. However, because of the explicit treatment of the acceleration terms, the time step necessary for stability is limited by the CFL condition in Equation 2-183.