1D RKDG to shallow water equations

RKDG to shallow water equations

1.Governing Equations

\[\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} = 0
\]

\[U = \begin{bmatrix} h \cr q \end{bmatrix} \quad
F = \begin{bmatrix} q \cr gh^2/2 + q^2/h \end{bmatrix}\]

2.Discrete with DGM

\[\begin{equation} U_h = \sum{l_j U_j} \quad F_h(U) = \sum{l_j F(U_j)} \end{equation}
\]

\[\begin{equation}\int_{\Omega} l_i l_j \frac{\partial U_j}{\partial t} dx+
\int_{\Omega} l_i \frac{\partial l_j}{\partial x} F(U_j) dx= 0 \end{equation}\]

\[\begin{equation} \int_{\Omega} l_i l_j \frac{\partial U_j}{\partial t} dx +
\int_{\Omega} l_i \frac{\partial l_j}{\partial x} F(U_j) dx+
\oint_{\partial \Omega} l_i l_j (F^* - F)\cdot \vec{n} ds = 0 \end{equation}\]

\[\begin{equation} JM \frac{\partial U}{\partial t} + JMD_x F(U) + J_E M_E (F^* - F)\cdot \vec{n} = 0 \end{equation}
\]

ODE:

\[\begin{equation} \frac{\partial U}{\partial t} = -\frac{\partial r}{\partial x}D_r F(U) + \frac{J_E}{J}M^{-1} M_E (F^* - F)\cdot \vec{n}=L(U(t)) \end{equation}
\]

\[\begin{equation} rhs = -\frac{\partial r}{\partial x}D_r F(U) + \frac{J_E}{J}M^{-1} M_E (F - F^*)\cdot \vec{n}\end{equation}
\]

It is important to point out that at dry cells no flux is flow inside the elemnt. Therefor, for dry cells

\[\begin{equation} rhs = \frac{J_E}{J}M^{-1} M_E (F - F^*)\cdot \vec{n}\end{equation}
\]

3.Numerical Flux

3.1.HLL flux function

Formulations are given as

\[F^{HLL} = \left\{ \begin{matrix}
F^- \cr
\frac{S_R F^- - S_L F^+ + S_L S_R(U^+ - U^-)}{S_R S_L} \cr
F^+ \end{matrix} \right.
\begin{matrix}
S_L \geq 0 \cr
S_L < 0 < S_R \cr
S_R \leq 0
\end{matrix}\]

Wave Speed is suggested by Fraccarollo and Toro (1995)

\[S_L = min(u^- - \sqrt{gh^-}, u^* - c^*)
\]

\[S_R = min(u^+ + \sqrt{gh^+}, u^* + c^*)
\]

\(u^*\) and \(c^*\) is defined by

\[u^* = \frac{1}{2}(u^- + u^+) + \sqrt{gh^-} - \sqrt{gh^+}
\]

\[c^* = \frac{1}{2}(\sqrt{gh^-} + \sqrt{gh^+}) + \frac{1}{4}(u^- - u^+)
\]

for wet-dry interface, the wave speed is giving as

1. left-hand dry bed

\[\begin{equation}
S_L = u^+ - 2\sqrt{g h^+} \quad S_R = u^+ + \sqrt{g h^+}
\end{equation}\]

1. right-hand dry bed

\[\begin{equation}
S_L = u^- - \sqrt{g h^-} \quad S_R = u^- + 2\sqrt{g h^-}
\end{equation}\]

1. both sides are dry

\[\begin{equation}
S_L = 0 \quad S_R = 0
\end{equation}\]

Noticing. 1

For flux terms, the discharge \(q^2\) is divided by water depth \(h\)

\[F = \begin{bmatrix} q \cr gh^2/2 + q^2/h \end{bmatrix}
\]

so a threadhold of water depth \(h_{flux}\) ( \(10^{-3}\)m ) is add into flux function SWEFlux.m. When \(h\) is less than \(h_{flux}\), the \(q^2/h\) is approximated to 0 as there is no flow at this node.

Noticing. 2

When defining the dry beds, another threadhold of water depth \(h_{dry}\) is used. It is convenient to deine \(h_{dry}\) equals to \(h_{flux}\).

3.2.Rotational invariance

\[T = \begin{bmatrix} 1 & 0 \cr
0 & n_x\end{bmatrix} \quad
T^{-1} = \begin{bmatrix} 1 & 0 \cr
0 & n_x\end{bmatrix}\]

\[\mathbf{F} \cdot \mathbf{n} = \mathbf{F} \cdot n_x = T^{-1}\mathbf{F}(TU)
\]

defining \(Q = TU\), the numerical flux \(\hat{\mathbf{F}}\) can be obtained through the evaluation of numerical flux \(\mathbf{F}\) by

\[\hat{\mathbf{F}} \cdot n = T^{-1}\mathbf{F}^{HLL}(Q)
\]

4.Limiter

Note: discontinuity detector from Krivodonova (2003) is not working

For better numerical stability, minmod limiter is used in limiting the discharge and elevation.

Check testing/Limiter1D/doc for more details about minmod limiter.

5. Positive preserving limiter

For the thin layer approach, a small depth ( \(h_{positive} = 10^{-3} m\)) and zeros velocity are prescribed for dry nodes.

The first step is to define wet elements. After each time step, the whole domain is calculated; If the any depth of nodes in \(\Omega_i\) is greater than \(h_{positive}\), then the element is defined as wet element, otherwise the water height of all nodes are remain unchanged.

The second step is to modify wet cells; If the depth of any nodes is less than \(h_{positive}\), then the flow rate is reset to zero and the new water depth is constructed as

\[\begin{equation}
\mathrm{M}\Pi_h h_i(x) = \theta_1 \left( h_i(x) - \bar{h}_i \right) + \bar{h}_i
\end{equation}\]

where

\[\begin{equation}
\theta_1 = min \left\{ \frac{\bar{h}_i - \xi }{\bar{h}_i - h_{min}}, 1 \right\}, \quad h_{min} = min\{ h_i (x_i) \}
\end{equation}\]

It is necessary to fulfill the restriction that the mean depth \(\bar{h}_i\) is greater than \(\xi\), i.e. \(10^{-4}\)m. In the function PositiveOperator, if the mean depth of element is less than \(\xi\), all nodes will add a small depth \(\xi - \bar{h}_i\) to re-fulfill the restriction.

At last, all values of water height at nodes with negative \(h_i(x_j) <0\) will be modified to zero and the discharge of dry nodes ( \(h_i \le h_{positive}\) ) will be reseted to zero.

6. Wet/Dry reconstruction

No special treatment is introduced in the model at the moment.

5.Numerical Test

5.1.Wet dam break

Model Setting	value
channel length	1000m
dam position	500m
upstream depth	10m
downstream depth	2m
element num	400
Final Time	20s

5.2.Dry dam break

Model Setting	value
channel length	1000m
dam position	500m
upstream depth	10m
downstream depth	0m
element num	400
Final Time	20s

5.3.Parabolic bowl

Model Setting	value
channel length	2000m
\(h_0\)	10m
\(a\)	600m
\(B\)	5m/s
\(T\)	269s

Exact solution

\[\begin{equation}
Z(x,t) = \frac{-B^2 \mathrm{cos}(2wt) - B^2 - 4Bw \mathrm{cos}(wt)x}{4g}
\end{equation}\]

1. \(t = T/2\)
   
   

2. \(t = 3T/4\)
   
   

3. \(t = T\)