An Entropy-Stable Residual Distribution Scheme for the System of Two-Dimensional Inviscid Shallow Water Equations

Abstract

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.

Appendices

Median-Dual Area of FV Method

To account for highly skewed cells which constitutes of obtuse triangles, Voronoi cells are replaced by median-dual cells in this study of the effect of grid randomization. By joining the centroids to the midpoints of the edges to form the boundary, median-dual cells are well defined in any triangulation as shown in Fig. 15a (cf. Fig. 1a).

Each edge in the Voronoi cells that separates the two adjacent points are further split into two shorter edges in the median-dual cells, e.g. edge i is split into its component in cell I and cell VI. Consequently, the flux between nodes 0 and 1 is split into $\mathbf {{\varvec{f}}}_{i,I}$ and $\mathbf {{\varvec{f}}}_{i,VI}$ with their respective normals, ${\hat{n}}_{i,I}$ and ${\hat{n}}_{i,VI}$, as demonstrated in Fig. 15b (cf. Fig. 1b). While having more edges with different normals, the fluxes are still computed as functions from variables from only the two adjacent nodes. In another words, the underlying lack of multidimensional property of FV method remains even median-dual cells are used.

Fig. 15

The first-order finite volume cell vertex diagram

Entropy-Stable FV Formulation

Following the work of [22] on Euler, the entropy-stable FV for SWE can be derived in a similar approach. The FV formulation starts with the semi-discretized SWE on arbitrary grids in 2D, and it is in the form of Eq. (12). With the definition of the length of a particular edge as $l_e=\sqrt{(\Delta x)^2_e+(\Delta y)^2_e}$ as illustrated in Fig. 16, the normal and tangential velocities at e are given by

$$\begin{aligned} q_e&=\frac{u\Delta y-v\Delta x}{l_e} \end{aligned}$$

(44)

$$\begin{aligned} r_e&=\frac{u\Delta x+v\Delta y}{l_e}. \end{aligned}$$

(45)

Fig. 16

A sample median-dual edge with normal and tangential velocities (q, r) in an arbitrary grid

Substituting the new definitions into Eq. (12), the normal flux through the edge will be

$$\begin{aligned} {\varvec{f}}_e n_{e,x} + {\varvec{g}}_e n_{e,y} = \left[ \begin{array}{c} h q\\ h u q + \frac{g h^2}{2}\frac{\Delta y}{l_e}\\ h v q - \frac{g h^2}{2}\frac{\Delta x}{l_e} \end{array} \right] l_e = \mathbf {{\varvec{F}}}_e l_e, \end{aligned}$$

(46)

We consider the two neighbouring points’ variables separated by the common median-dual edge to be subscripted by L and R. The net flux across edge e is the summation of the symmetric entropy conserving flux, ${\varvec{F}}_C$, and an entropy-stable (or consistent) dissipative flux, which is given by

$$\begin{aligned} \mathbf {{\varvec{F}}_e} ({\varvec{u}}_L,{\varvec{u}}_R) = {\varvec{F}}_C ({\varvec{u}}_L,{\varvec{u}}_R) - \frac{1}{2}\hat{{\varvec{R}}}|\hat{\varvec{\varLambda }}\hat{{\varvec{S}}}|\hat{{\varvec{R}}}^{{\varvec{T}}} [{\varvec{v}}]. \end{aligned}$$

(47)

The entropy conservative flux is computed with arithmetic averaged quantities from the neighbouring two points, and it is defined as

$$\begin{aligned} {\varvec{F}}_C ({\varvec{u}}_L,{\varvec{u}}_R) = \left[ \begin{array}{c} {\hat{h}} {\hat{q}}\\ {\hat{h}} {\hat{u}} {\hat{q}} + \frac{g {\hat{h}}^2}{2}\frac{\Delta y}{l_e}\\ {\hat{h}} {\hat{v}} {\hat{q}} - \frac{g {\hat{h}}^2}{2}\frac{\Delta x}{l_e} \end{array} \right] \end{aligned}$$

(48)

where ${\hat{a}}=\frac{1}{2}\left( a_L + a_R \right)$. It should be noted that the right eigenvectors, $\hat{{\varvec{R}}}$, the eigenvalues, $\hat{\varvec{\varLambda }}$, and the scaling matrix, $\hat{{\varvec{S}}}$, in the dissipative flux are of similar forms as Eqs. (25), (26), (27), respectively. However, the components of these matrices in Eq. (47) are also arithmetic averaged values of the two points. Besides, the streamline direction $\theta$ is replaced by the edge outward normal, $\hat{n_e}$. $[{\varvec{v}}]$ is defined as the change of the entropy variables between the left and right points, $[{\varvec{v}}]={\varvec{v}}_R-{\varvec{v}}_L$.

Edge-Based Fluxes

Consider any system of hyperbolic equations as given in Eq. (1) with the fluxes in the x-direction given as ${\varvec{f}}$. We intend to determine the difference between the flux over the element ${\varvec{f}}^*$ and the arithmetic average flux over the edge of the particular element ${\varvec{f}}^{*}_{ij},{\varvec{f}}^{*}_{jk},{\varvec{f}}^{*}_{ki}$ as shown in Fig. 4. Assume that this particular element has ${\varvec{f}}^*$ located at (0, 0). Thus, the centre of the triangle is at the (0, 0) or,

$$\begin{aligned} x_i+x_j+x_k=0,\qquad y_i+y_j+y_k=0 \end{aligned}$$

(49)

and the coordinates are,

$$\begin{aligned} i:(x_ih,y_ih),\quad j:(x_jh,y_jh),\quad k:(x_kh,y_kh) \end{aligned}$$

(50)

The three entropy-conserved fluxes can be calculated for each edge

$$\begin{aligned} {\varvec{f}}^*_{ij},\quad {\varvec{f}}^*_{jk},\quad {\varvec{f}}^*_{ki} \end{aligned}$$

(51)

as given in Eq. (18). Using Taylor analysis, we can estimate the error as

$$\begin{aligned} \begin{array}{rl} \text {Error}=&{}\displaystyle {\varvec{f}}^*- \frac{{\varvec{f}}^*_{ij}+{\varvec{f}}^*_{jk}+{\varvec{f}}^*_{ki}}{3} \\ =&{}\displaystyle \frac{h^2}{3}\left( -\frac{\partial ^2{\varvec{f}}^*}{\partial x^2}\left( x_i^2+x_ix_j+x_j^2\right) \right. \\ &{}\displaystyle -\frac{\partial ^2{\varvec{f}}^*}{\partial x\partial y} \left( -x_i (2 y_i + y_j) - x_j (y_i + 2 y_j)\right) \\ &{}\displaystyle \left. -\frac{\partial ^2{\varvec{f}}^*}{\partial y^2} \left( y_i^2+y_iy_j+y_j^2\right) \right) +O\left( h^3\right) \end{array} \end{aligned}$$

(52)

which demonstrates error of $O(h^2)$.

Classic RD Schemes

In the context of RD methods, there are various approaches to distribute the total residual or signals to each node of the element. Considering system of conservation laws in Eq. (1), the upwind parameter for a certain node i in an element is introduced as

$$\begin{aligned} {\varvec{k}}_i=\frac{1}{2}\left( \overrightarrow{{\varvec{R}}}\overrightarrow{\varvec{\varLambda }}\overrightarrow{{\varvec{R}}}^{-1}\right) \cdot \overrightarrow{n}_i, \end{aligned}$$

(53)

where $\overrightarrow{n}_i$ is the inward normal of the edge opposite of node i scaled with the respective edge. To differentiate the fluxes flowing in or out of the cell, define

$$\begin{aligned} {\varvec{k}}_i^+=\left\{ \begin{array}{lc} {\varvec{k}}_i&{}\quad \left( \overrightarrow{{\varvec{R}}}\overrightarrow{\varvec{\varLambda }}\overrightarrow{{\varvec{R}}}^{-1}\right) \cdot \overrightarrow{n}_i\ge 0 \\ 0&{}\quad \left( \overrightarrow{{\varvec{R}}}\overrightarrow{\varvec{\varLambda }}\overrightarrow{{\varvec{R}}}^{-1}\right) \cdot \overrightarrow{n}_i<0 \end{array} \right. ,\qquad {\varvec{k}}_i^-=\left\{ \begin{array}{lc} 0&{}\quad \left( \overrightarrow{{\varvec{R}}}\overrightarrow{\varvec{\varLambda }}\overrightarrow{{\varvec{R}}}^{-1}\right) \cdot \overrightarrow{n}_i>0 \\ {\varvec{k}}_i&{}\quad \left( \overrightarrow{{\varvec{R}}}\overrightarrow{\varvec{\varLambda }}\overrightarrow{{\varvec{R}}}^{-1}\right) \cdot \overrightarrow{n}_i\le 0 \end{array} \right. \end{aligned}$$

(54)

where ${\varvec{k}}_i^+$ represents flux flowing into the element at edge opposite of node i and on the same note, ${\varvec{k}}_i^-$ represents outflowing flux. Then nodal signal distributions for the classic RD schemes are defined as the following.

• Narrow (N) scheme

  $$\begin{aligned} \varvec{\phi }_i^\text {N}={\varvec{k}}_i^+\left( {\varvec{u}}_i-\hat{{\varvec{u}}}\right) ,\quad \hat{{\varvec{u}}}=\left( \sum {\varvec{k}}_i^-\right) ^{-1}\left( \sum {\varvec{k}}_i^-{\varvec{u}}_i\right) \end{aligned}$$

  (55)

• Lax–Friedrichs (LxF) scheme

  $$\begin{aligned} \varvec{\phi }_i^{\mathrm{LxF}}=\bar{\varvec{\phi }}_T+{\varvec{a}}({\varvec{u}}_i-\bar{{\varvec{u}}}),\quad {\varvec{a}}\ge \max (|{\varvec{k}}_i|,|{\varvec{k}}_j|,|{\varvec{k}}_k|) \end{aligned}$$

  (56)

  where $\bar{{\varvec{u}}}$ and $\bar{\varvec{\phi }}_T$ are the arithmetic average of ${\varvec{u}}$ and $\varvec{\phi }_T$ over the element.

• Low diffusion advection (LDA) scheme

  $$\begin{aligned} \varvec{\phi }_i^\text {LDA}=\varvec{\zeta }_i^{LDA}\varvec{\phi }_T,\quad \varvec{\zeta }_i^\text {LDA}={\varvec{k}}_i^+\left( \sum {\varvec{k}}_i^+\right) ^{-1} \end{aligned}$$

  (57)

• Lax–Wendroff (LxW) scheme

  $$\begin{aligned} \varvec{\phi }_i^{\mathrm{LxW}}=\varvec{\zeta }_i\varvec{\phi }_T,\quad \varvec{\zeta }_i=\frac{1}{3}+\frac{\Delta t}{2A_i}{\varvec{k}}_i,\quad \Delta t=\left( \frac{2A_i}{\sum _p|{\varvec{k}}_p|}\right) \nu . \end{aligned}$$

  (58)

  where $A_i$ is the median-dual area of node i and $\nu$ is the CFL number.