Hydrostatic reconstruction with a dry-safe HLL flux and one-sided pressure corrections

Let $\mathrm{\Delta }x>0$ be a uniform cell width, let $i$ range over a finite chain or a periodic or bi-infinite mesh, and let $b_i$ be the bed value in cell $i$. At time level $n$, the cell state is $U_i^n=(h_i^n,m_i^n)\in K$ from One-dimensional shallow-water balance law with a dry-state convention, with velocity $u_i^n$ and free surface ${\eta }_i^n=h_i^n+b_i$.

For a finite chain $i=1,\dots ,N$ with $N\ge 1$, supply at each time level two ghost states $U_0^n,U_{N+1}^n\in K$ and bed values $b_0,b_{N+1}\in \mathbb{R}$. These are boundary inputs, used only to evaluate the faces $1\mathrm{/}2$ and $N+1\mathrm{/}2$; only cells $1,\dots ,N$ are updated. On a periodic mesh, identify neighboring indices periodically.

At face $i+1\mathrm{/}2$, set $b_{i+1\mathrm{/}2}^{\ast }=\max (b_i,b_{i+1}),h_{i+1\mathrm{/}2,L}^{\ast }=\max (0,{\eta }_i^n-b_{i+1\mathrm{/}2}^{\ast }),h_{i+1\mathrm{/}2,R}^{\ast }=\max (0,{\eta }_{i+1}^n-b_{i+1\mathrm{/}2}^{\ast })\mathrm{.}$ For $s=L,R$, set $u_{i+1\mathrm{/}2,s}^{\ast }=u_i^n$ or $u_{i+1}^n$, respectively, when $h_{i+1\mathrm{/}2,s}^{\ast }>0$, and set it to zero when $h_{i+1\mathrm{/}2,s}^{\ast }=0$. Define $U_{i+1\mathrm{/}2,s}^{\ast }=(h_{i+1\mathrm{/}2,s}^{\ast },h_{i+1\mathrm{/}2,s}^{\ast }{u}_{i+1\mathrm{/}2,s}^{\ast })$ and $c_{i+1\mathrm{/}2,s}^{\ast }=\sqrt{g{h}_{i+1\mathrm{/}2,s}^{\ast }}$. The face speeds are $a_{L,i+1\mathrm{/}2}=\min (0,u_{i+1\mathrm{/}2,L}^{\ast }-c_{i+1\mathrm{/}2,L}^{\ast },u_{i+1\mathrm{/}2,R}^{\ast }-c_{i+1\mathrm{/}2,R}^{\ast }),a_{R,i+1\mathrm{/}2}=\max (0,u_{i+1\mathrm{/}2,L}^{\ast }+c_{i+1\mathrm{/}2,L}^{\ast },u_{i+1\mathrm{/}2,R}^{\ast }+c_{i+1\mathrm{/}2,R}^{\ast })\mathrm{.}$ They satisfy $a_L\le 0\le a_R$. Write $F_s^{\ast }=F(U_{i+1\mathrm{/}2,s}^{\ast })$. The common HLL face flux is ${\widehat{F}}_{i+1\mathrm{/}2}^n=\{\begin{array}{ll}{F}_L^{\ast },& a_{L,i+1\mathrm{/}2}\ge 0,\\ F_R^{\ast },& a_{R,i+1\mathrm{/}2}\le 0,\\ {\displaystyle \frac{a_{R,i+1\mathrm{/}2}{F}_L^{\ast }-a_{L,i+1\mathrm{/}2}{F}_R^{\ast }+a_{L,i+1\mathrm{/}2}{a}_{R,i+1\mathrm{/}2}(U_{i+1\mathrm{/}2,R}^{\ast }-U_{i+1\mathrm{/}2,L}^{\ast })}{a_{R,i+1\mathrm{/}2}-a_{L,i+1\mathrm{/}2}}},& a_{L,i+1\mathrm{/}2}<0<a_{R,i+1\mathrm{/}2}\mathrm{.}\end{array}$ If both speeds are zero, both reconstructed states are dry and the first branch returns $(0,0)$; the middle denominator is never used in that case.

The one-sided corrected fluxes, where $L$ is the side of cell $i$ and $R$ the side of cell $i+1$, are $G_{i+1\mathrm{/}2,L}^n={\widehat{F}}_{i+1\mathrm{/}2}^n+(0,\frac{g}{2}((h_i^n{)}^2-(h_{i+1\mathrm{/}2,L}^{\ast }{)}^2\left)\right),G_{i+1\mathrm{/}2,R}^n={\widehat{F}}_{i+1\mathrm{/}2}^n+(0,\frac{g}{2}((h_{i+1}^n{)}^2-(h_{i+1\mathrm{/}2,R}^{\ast }{)}^2\left)\right)\mathrm{.}$ For $\mathrm{\Delta }t>0$, put $\lambda =\mathrm{\Delta }t\mathrm{/}\mathrm{\Delta }x$ and update $U_i^{n+1}=U_i^n-\lambda (G_{i+1\mathrm{/}2,L}^n-G_{i-1\mathrm{/}2,R}^n)\mathrm{.}$ Equivalently, with $h_{i,+}^{\ast }=h_{i+1\mathrm{/}2,L}^{\ast }$, $h_{i,-}^{\ast }=h_{i-1\mathrm{/}2,R}^{\ast }$, and $\widehat{F}$ the shared face flux, $U_i^{n+1}=U_i^n-\lambda ({\widehat{F}}_{i+1\mathrm{/}2}^n-{\widehat{F}}_{i-1\mathrm{/}2}^n)+\mathrm{\Delta }t{S}_i^n,S_i^n=(0,\frac{g}{2\mathrm{\Delta }x}((h_{i,+}^{\ast }{)}^2-(h_{i,-}^{\ast }{)}^2\left)\right)\mathrm{.}$ The face-specific depth CFL condition is $\lambda (a_{R,i+1\mathrm{/}2}-a_{L,i-1\mathrm{/}2})\le 1\text{for every cell }i\mathrm{.}$

The diagram is a wet/wet face with $b_i<b_{i+1}$ and a common rest level ${\eta }_0$. The dotted line marks the free surface, the solid step is the bed, and both reconstructed depths are measured from $b^{\ast }=\max (b_i,b_{i+1})$. The displayed equations specify every state and orientation used in proofs.

diagram failed to render:
\begin{tikzpicture}[x=1.1cm,y=1cm]
  \draw[->] (0,0) -- (4.5,0) node[right] {$x$};
  \draw[thick] (0,0.35) -- (2,0.35) -- (2,1.0) -- (4,1.0);
  \draw[dotted,thick] (0,1.8) -- (4,1.8) node[right] {$\eta_0$};
  \draw[dashed] (2,1.0) -- (4,1.0);
  \draw[<->] (1,1.0) -- (1,1.8) node[midway,left] {$h_L^*$};
  \draw[<->] (3,1.0) -- (3,1.8) node[midway,right] {$h_R^*$};
  \draw (2,0.1) -- (2,1.95);
  \node at (1,0.12) {$b_i$};
  \node at (3,0.75) {$b_{i+1}=b^*$};
  \node at (2,2.08) {$i+\tfrac12$};
\end{tikzpicture}

The scheme uses only the cell bed values, so its discrete fluxes are defined for bed steps. A continuum source or weak-limit statement across a discontinuous bed requires a separately specified path product and is not supplied here. No entropy inequality or stiff-relaxation/asymptotic-preserving claim is part of this definition.