Well-balanced schemes for the shallow water equations with Coriolis forces

Abstract

In the present paper we study shallow water equations with bottom topography and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. In order to construct a higher order approximation a crucial step is a well-balanced reconstruction which has to be combined with a well-balanced update in time. We implement our newly developed second-order reconstruction in the context of well-balanced central-upwind and finite-volume evolution Galerkin schemes. Theoretical proofs and numerical experiments clearly demonstrate that the resulting finite-volume methods preserve exactly the so-called jets in the rotational frame. For general two-dimensional geostrophic equilibria the well-balanced methods, while not preserving the equilibria exactly, yield better resolution than their non-well-balanced counterparts.

A derivation of FVEG evolution operators

In this section, we derive new well-balanced evolution operators $E_\tau ^{const}$ (4.11) and $E_\tau ^{bilin}$ (4.14), used in (4.2), (4.3) and (4.4) with $\tau =\Delta t/2$. Here, we only show how to evolve the values of the potentials K and L, since the evolution equations for the velocities u and v have been presented in [36].

The solution is going to be evolved from time $t^n$ to time $t^n+\tau$ and we are going to use the following notations, which are similar to the notations that were used in Sect. 4:

$$\begin{aligned} \begin{aligned}&P{:}{=}(x,y,t^n+\tau ),\quad Q_0(t){:}{=}\big (x-u^*(t^n+\tau -t),\, y-v^*(t^n+\tau -t),\,t\big ),\\&Q(t){:}{=}\big (x-[u^*-c^*\cos \theta ](t^n+\tau -t),\,y-[v^*-c^*\sin \theta ](t^n+\tau -t),\,t),\\&\quad t\in [t^n,t^n+\tau ), \end{aligned} \end{aligned}$$

where, as before, $\theta \in (0,2\pi ]$, and $u^*$, $v^*$ and $c^*$ are the local velocities and speed of gravity waves at the point $(x,y,t^n)$. Note that P is the vertex of the bicharacteristic cone, $Q_0(t)$ are the points along the cone axis, Q(t) are the points on the mantle of the cone, and, in particular, $Q(t^n)$ are the points at the perimeter of the sonic circle at time $t^n$.

In order to obtain an expression for the evolution operators for K and L, we first write the exact integral equation for water depth h, which can be derived using the theory of bicharacteristics, see [36, (A5)]:

$$\begin{aligned} h(P)= & {} \frac{1}{2\pi }\int \limits _0^{2\pi }\left[ h(Q(t^n)) -\frac{c^*}{g}\Big (u(Q(t^n))\cos \theta -v(Q(t^n))\sin \theta \Big )\right] d\theta \nonumber \\&-\frac{1}{2\pi }\int \limits _{t^n}^{t^n+\tau }\bigg [\frac{1}{t^n +\tau -t}\int \limits _0^{2\pi }\frac{c^*}{g}\Big (u(Q(t))\cos \theta + v(Q(t))\sin \theta \Big )d\theta \bigg ]dt\nonumber \\&+\frac{c^*}{2\pi }\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [B_x(Q(t))\cos \theta +B_y(Q(t))\sin \theta \Big ]d\theta dt\nonumber \\&-\frac{c^*}{2\pi }\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [V_x(Q(t))\cos \theta -U_y(Q(t))\sin \theta \Big ]d\theta dt. \end{aligned}$$

(A.1)

We now rewrite the last integral on the RHS of (A.1) as

$$\begin{aligned} \begin{aligned}&\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [V_x(Q(t))\cos \theta -U_y(Q(t))\sin \theta \Big ]d\theta dt\\&=\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [V_x(Q(t))\cos \theta +V_y(Q(t))\sin \theta \Big ]d\theta dt\\&\quad - \int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi }\Big [V_y(Q(t)) \cos \theta +U_y(Q(t))\sin \theta \Big ]d\theta dt \end{aligned} \end{aligned}$$

(A.2)

Applying the rectangle rule for time integration and the Taylor expansion about $Q_0(t^n)$, we can show that the last integral in (A.2) is of order $\mathcal{O}(\tau ^2)$:

$$\begin{aligned} \begin{aligned}&\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [V_y(Q(t))\cos \theta +U_y(Q(t))\sin \theta \Big ]d\theta dt\\&=\int \limits _0^{2\pi }\Big [V_y(Q(t^n))\cos \theta +U_y(Q(t^n)) \sin \theta \Big ]d\theta +\mathcal{O}(\tau ^2)\\&=V_y(Q_0(t^n))\int \limits _0^{2\pi }\cos \theta \,d\theta +U_y(Q_0(t^n))\int \limits _0^{2\pi }\sin \theta \,d\theta +\mathcal{O}(\tau ^2)= \mathcal{O}(\tau ^2). \end{aligned} \end{aligned}$$

To evaluate the first integral on the RHS of (A.2), we introduce polar-type coordinates along the mantle of the bicharacteristic cone

$$\begin{aligned} \xi =x+r\Big (\cos \theta -\frac{u^*}{c^*}\Big ),\quad \eta =y+r\Big (\sin \theta -\frac{v^*}{c^*}\Big ), \end{aligned}$$

where $r=c^*(t^n+\tau -t)$ is the circle radius at time level $t\in [t^n,t^n+\tau ]$. Thus, we have

$$\begin{aligned} \frac{dV}{dr}(r,\theta )=V_x(\xi ,\eta )\cos \theta +V_y(\xi ,\eta )\sin \theta -\frac{1}{c^*}\Big (u^*V_x(\xi ,\eta ) +v^*V_y(\xi ,\eta )-V_t(\xi ,\eta ) \Big ), \end{aligned}$$

and hence we obtain

$$\begin{aligned} \begin{aligned}&\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [V_x(Q(t))\cos \theta +V_y(Q(t))\sin \theta \Big ]d\theta dt\\&=-\frac{1}{c^*}\int \limits _{c^*\tau }^0\int \limits _0^{2\pi } \frac{dV(r,\theta )}{dr}\,d\theta dr\\&\quad + \frac{1}{c^*}\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*V_x(Q(t))+v^*V_y(Q(t))+V_t(Q(t))\Big ]d\theta dt\\&=\frac{1}{c^*}\int \limits _0^{c^*\tau }\frac{d}{dr}\Bigg (\int \limits _0^{2\pi }V(r,\theta )\,d\theta \Bigg )dr\\&\quad + \frac{1}{c^*}\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*V_x(Q(t))+v^*V_y(Q(t))+V_t(Q(t))\Big ]d\theta dt\\&=\frac{1}{c^*}\bigg (\int \limits _0^{2\pi }V(Q(t^n))\,d\theta -2\pi V(P)\bigg )\\&\quad + \frac{1}{c^*}\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*V_x(Q(t))+v^*V_y(Q(t))+V_t(Q(t))\Big ]d\theta dt. \end{aligned} \end{aligned}$$

(A.3)

Similarly, the third integral on the RHS of (A.1) is equal to

$$\begin{aligned} \begin{aligned}&\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [B_x(Q(t))\cos \theta +B_y(Q(t))\sin \theta \Big ]d\theta dt\\&=\frac{1}{c^*}\bigg (\int \limits _0^{2\pi }B(Q(t^n))\,d\theta -2\pi B(P)\bigg )+ \frac{1}{c^*}\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*B_x(Q(t))+v^*B_y(Q(t))\Big ]d\theta dt, \end{aligned} \end{aligned}$$

(A.4)

since $B_t\equiv 0$.

Combining (A.1), (A.3) and (A.4), we obtain the following approximation of the exact integral equations for $K=g(h+B-V)$:

$$\begin{aligned} K(P)= & {} \frac{1}{2\pi }\int \limits _0^{2\pi }\bigg [K(Q(t^n)) -c^*\Big (u(Q(t^n))\cos \theta +v(Q(t^n))\sin \theta \Big )\bigg ]d\theta \nonumber \\&-\frac{1}{2\pi }\int \limits _{t^n}^{t^n+\tau }\bigg [\frac{1}{t^n+\tau -t}\int \limits _0^{2\pi }c^*\Big (u(Q(t))\cos \theta + v(Q(t))\sin \theta \Big )d\theta \bigg ]dt\nonumber \\&+\frac{g}{2\pi }\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*B_x(Q(t))+v^*B_y(Q(t))\Big ]d\theta dt\nonumber \\&-\frac{g}{2\pi }\int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*V_x(Q(t))+v^*V_y(Q(t))+V_t(Q(t))\Big ]d\theta dt. \end{aligned}$$

(A.5)

This formula provides the evolution equation for the equilibrium variable K. Our next goal is to approximate time integrals in (A.5) in a suitable way, so that we obtain explicit approximate evolution operators for $E_\tau ^{const}$ (4.11) and $E_\tau ^{bilin}$ (4.14). This will be realized in a standard way following [35] and a superscript I will be used for piecewise constant approximations, while a superscript II will be used for piecewise linear ones.

For the piecewise constant approximations, the spatial derivatives are zero and thus the corresponding integral terms in (A.5) vanish. Therefore, using the fact that

$$\begin{aligned} \int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi }V^\mathrm{I}_t(Q(t))\, d\theta dt= 2\pi \big [V^\mathrm{I}(P)-V^\mathrm{I}(Q_0(t^n))\big ]+\mathcal{O}(\tau ^2) \end{aligned}$$

(A.6)

and approximating the mantle integral of $\,u^\mathrm{I}(Q(t))\cos \theta +v^\mathrm{I}(Q(t))\sin \theta \,$ according to [35], we obtain

$$\begin{aligned} \begin{aligned} K^\mathrm{I}(P)&\approx \frac{1}{2\pi }\int \limits _0^{2\pi } \Big [K^\mathrm{I}(Q(t^n))-c^*\Big (u^\mathrm{I}(Q(t^n))\text{ sgn }(\cos \theta )+ v^\mathrm{I}(Q(t^n))\text{ sgn }(\sin \theta )\Big )\Big ]d\theta \\&\quad +g\big [V^\mathrm{I}(Q_0(t^n))-V^\mathrm{I}(P)\big ], \end{aligned} \end{aligned}$$

which leads to (4.11).

For the piecewise linear approximations, the spatial derivatives do not vanish and we approximate the integrals in the corresponding terms in (A.5) by the rectangle rule to obtain

$$\begin{aligned} \int \limits _{t^n}^{t^n+\tau }\int \limits _0^{2\pi } \Big [u^*V^\mathrm{II}_x(Q(t))+v^*V^\mathrm{II}_y(Q(t))\Big ] d\theta dt= \tau \int \limits _0^{2\pi }\Big [u^*V^\mathrm{II}_x(Q(t^n)) +v^*V^\mathrm{II}_y(Q(t^n))\Big ]d\theta +\mathcal{O}(\tau ^2), \end{aligned}$$

which, together with a similar approximation of the bottom topography terms, leads to (4.14). Indeed, applying (A.6) and the standard approximation for the mantle integral of $\,u^\mathrm{II}(Q(t))\cos \theta +v^\mathrm{II}(Q(t))\sin \theta \,$ as in [35], we finally obtain

$$\begin{aligned} \begin{aligned}&K^\mathrm{II}(P)=K^\mathrm{II}(Q_0(t^n))+\frac{1}{4}\int \limits _0^{2\pi }\Big (K^\mathrm{II}(Q(t^n))-K^\mathrm{II} (Q_0(t^n))\Big )d\theta \\&\quad -\frac{c^*}{\pi }\int \limits _0^{2\pi }\Big (u^\mathrm{II}(Q(t^n)) \cos \theta +v^\mathrm{II}(Q(t^n))\sin \theta \Big )d\theta \\&\quad +\frac{\tau }{2\pi }\int \limits _0^{2\pi }\Big [u^*\big (K^\mathrm{II}_x(Q(t^n))-gh^\mathrm{II}_x(Q(t^n))\big ) +v^*\big (K^\mathrm{II}_y(Q(t^n))-gh^\mathrm{II}_y(Q(t^n))\big ) \Big ]d\theta \\&\quad +g\big [V^\mathrm{II}(Q_0(t^n))-V^\mathrm{II}(P)\big ]. \end{aligned} \end{aligned}$$

Notice that the derivation of the approximate evolution operators for $L^\mathrm{I}(P)$ and $L^\mathrm{II}(P)$ is analogous.