The 1D SWEs are obtained by assuming an incompressible fluid in a channel of unit width with negligible vertical velocity and almost constant horizontal velocity across any cross section of the channel. This assertion remains valid by considering small-amplitude waves in a fluid that is relatively shallow in comparison to its wavelength (see [3] ). In differential form, the conservative formulation of the homogeneous 1D SWEs is

${\partial }_{t}U+{\partial }_{x}F\left(U\right)=0,$ (2)

where $U$ and $F\left(U\right)$ are the vectors of conserved variables and fluxes, given respectively by

$U=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{c}h\\ hu\end{array}\right],F\left(U\right)=\left[\begin{array}{c}{f}_1\left(u_1,u_2\right)\\ f_2\left(u_1,u_2\right)\end{array}\right]=\left[\begin{array}{c}{u}_2\\ \frac{u_2^2}{u_1}+\frac{1}{2}g{u}_1^2\end{array}\right]=\left[\begin{array}{c}hu\\ h{u}^2+\frac{1}{2}g{h}^2\end{array}\right],$ (3)

where $u_i=u_i\left(x,t\right)$, $i=1,2$, are conserved variables, while $h=h\left(x,t\right)$ and $u=u\left(x,t\right)$ are primitive variables. These variables are functions of space $x$ and time $t$. Equations (2)-(3) express the physical laws of conservation of mass and momentum, where $h$ denotes the water depth, $u$ is the horizontal velocity of the fluid, and $g\approx 9.81\text{m}/{\text{s}}^{\text{2}}$ is the gravitational acceleration. In shallow water theory, the quantity $hu$ is frequently referred to as the discharge because it quantifies the water flow rate past a point.

For smooth solution, the SWEs (2)-(3) can alternatively be expressed in quasi-linear form

${\partial }_{t}U+A\left(U\right){\partial }_{x}U=0,$

where the coefficient matrix $A\left(U\right)$ is the Jacobian matrix

$\begin{array}{c}A\left(U\right)=\frac{\partial F}{\partial U}=\left[\begin{array}{cc}\frac{\partial f_1}{\partial u_1}& \frac{\partial f_1}{\partial u_2}\\ \frac{\partial f_2}{\partial u_1}& \frac{\partial f_2}{\partial u_2}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -{\left(\frac{u_2}{u_1}\right)}^2+g{u}_1& \frac{2u_2}{u_1}\end{array}\right]\\ =\left[\begin{array}{cc}0& 1\\ -u^2+gh& 2u\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -u^2+c^2& 2u\end{array}\right],\end{array}$ (4)

where $c=\sqrt{gh}$ represents the speed of a shallow water wave, which is often referred to as celerity. The eigenvalues of $A\left(U\right)$ are

${\lambda }_1=\frac{u_2}{u_1}-\sqrt{g{u}_1}=u-c,{\lambda }_2=\frac{u_2}{u_1}+\sqrt{g{u}_1}=u+c,$ (5)

with the corresponding right eigenvectors

$R^{\left(1\right)}=\left[\begin{array}{c}1\\ {\lambda }_1\end{array}\right]=\left[\begin{array}{c}1\\ u-c\end{array}\right],R^{\left(2\right)}=\left[\begin{array}{c}1\\ {\lambda }_2\end{array}\right]=\left[\begin{array}{c}1\\ u+c\end{array}\right].$ (6)

The matrix having columns that are eigenvectors (6) is denoted as

$R=R\left(u,c\right)=\left[\begin{array}{cc}1& 1\\ u-c& u+c\end{array}\right].$ (7)

Then, the rows of the subsequent matrix

$L=R^{-1}\left(u,c\right)=\left[\begin{array}{cc}\frac{u+c}{2c}& \frac{-1}{2c}\\ \frac{-u+c}{2c}& \frac{1}{2c}\end{array}\right],$ (8)

are left eigenvectors of $A\left(U\right)$.

The SWEs are hyperbolic, i.e., the System (2)-(3) of 2 equations has 2 real eigenvalues. If the celerity $c$ remains positive, the equations are strictly hyperbolic, meaning that the eigenvalues are real and distinct.

The subsequent notations are in accordance with those in [2] . The RP for (2)-(3) is the initial value problem for (2)-(3) with Initial Conditions (ICs):

$U\left(x,0\right)=\{\begin{array}{ll}{U}_L={\left[u_{1L},u_{2L}\right]}^{\top }\hfill & \text{if}x<x_0,\hfill \\ U_R={\left[u_{1R},u_{2R}\right]}^{\top }\hfill & \text{if}x>x_0,\hfill \end{array}$ (9)

and is represented by the notation RP ( $U_L$, $U_R$). Subscripts L and R indicate the left and right states, respectively, of the piecewise constant data with a single jump discontinuity at some point, say $x=x_0$. It is possible to solve the initial value problem (2), (3), (9) exactly. Figure A1 in Appendix A depicts the structure of the solution to the RP (2), (3), (9) in the x-t plane, in the manner indicated in [23] .

The solution is a similarity solution, i.e., a function only dependent on the ratio $x/t$. Two waves, one corresponding to each eigenvalue ${\lambda }_i$, $i=1,2$, partition it into three constant states. These waves propagate away from the point of the initial discontinuity at constant speeds. The solution to the left of the ${\lambda }_1$-wave is equal to the initial data $U_L$, whereas the solution to the right of the ${\lambda }_2$-wave is equal to $U_R$. The region bounded by the ${\lambda }_1$ and ${\lambda }_2$ waves is commonly referred to as the Star Region, and the denotation $U^{\text{*}}$ represents the solution in that area.

There are two types of waves: shock waves and rarefaction waves. Within the context of shallow water theory, shock and rarefaction waves are alternatively referred to as bore and depression waves, respectively. In the following discussion, shock waves and rarefaction waves shall be referred to as shocks and rarefactions, respectively. Figure A2 of Appendix A, also available in [23] , illustrates each of the four potential wave patterns. At any given time moment, the solution varies continuously with x via rarefactions, whereas it jumps discontinuously via shocks.

When solving the RP (2), (3), (9), the speeds of the head and tail of a left or right rarefaction will be designated as $S_{HL}$, $S_{TL}$, $S_{HR}$, and $S_{TR}$, respectively. Whereas the speeds of a left or right shock will be represented as $S_{SL}$ and $S_{SR}$, correspondingly. The formulas needed to compute these speeds are available in [1] . If rarefactions are present, the solution within the left or right rarefaction, denoted as $U_{Lfan}$ and $U_{Rfan}$, respectively, is given by

$U_{Lfan}\left(x,t\right)={\left[\frac{{\left(2c_L+u_L-S\right)}^2}{9g},\frac{\left(u_L+2S+2c_L\right){\left(2c_L+u_L-S\right)}^2}{27g}\right]}^{\top },$ (10)

$U_{Rfan}\left(x,t\right)={\left[\frac{{\left(2c_R-u_R+S\right)}^2}{9g},\frac{\left(u_R+2S-2c_R\right){\left(2c_R-u_R+S\right)}^2}{27g}\right]}^{\top },$ (11)

where $S=\left(x-x_0\right)/t$, $u_K=u_{2K}/u_{1K}$, $c_K=\sqrt{g{u}_{1K}}$, $K=L,R$. See references [1] - [4] for specifics regarding the RP solutions. For SWEs, the exact solution to the RP is provided in [1] [3] .

Consider the 1D HCLs (1). The computational domain is discretized into grid

cells, or FVs $I_i=\left[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right]$. In the x-t plane, assume a control volume $V=I_i\times \left[t^n,t^{n+1}\right]$ with dimensions $\text{Δ}x=x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}$ and $\text{Δ}t=t^{n+1}-t^n$. One approach is to formulate entirely discrete one-step schemes, while the alternative is

to develop semi-discrete schemes. The latter approach is employed to execute the WENO schemes.

By maintaining the time variable $t$ continuous while integrating (1) over the volume V, the subsequent system of Ordinary Differential Equations (ODEs) is derived:

$\frac{\text{d}}{\text{d}t}{\bar{U}}_i\left(t\right)=-\frac{1}{\text{Δ}x}\left(F_{i+\frac{1}{2}}-F_{i-\frac{1}{2}}\right),$

where ${\bar{U}}_i\left(t\right)$ represents the spatial-integral average of the solution $U$ in the

cell $I_i$ at time $t$, commonly known as the cell average, while $F_{i+\frac{1}{2}}$ represents the physical flux at the cell boundary, or cell interface $x=x_{i+\frac{1}{2}}$ and time $t$, that

is

${\bar{U}}_i\left(t\right)=\frac{1}{\text{Δ}x}{\displaystyle {\int }_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}U\left(x,t\right)\text{d}x},F_{i+\frac{1}{2}}=F\left(U\left(x_{i+\frac{1}{2}},t\right)\right),$ (12)

where $U\left(x_{i+\frac{1}{2}},t\right)$ is the solution of (1) at the cell interface $x_{i+\frac{1}{2}}$. In consideration

of the subsequent approximations, semi-discrete FV schemes may be formulated.

Initially, $F_{i+\frac{1}{2}}$ must be estimated due to the fact that $U\left(x_{i+\frac{1}{2}},t\right)$ varies with

time along each cell interface and the exact solution is unavailable. Hence, the FV numerical scheme is expressed as

$\frac{\text{d}}{\text{d}t}{\bar{U}}_i\left(t\right)=-\frac{1}{\text{Δ}x}\left({\hat{f}}_{i+\frac{1}{2}}-{\hat{f}}_{i-\frac{1}{2}}\right)=L_i\left(U\right),$ (13)

where ${\hat{f}}_{i+\frac{1}{2}}$ is an approximation to $F_{i+\frac{1}{2}}$, known as the numerical flux or the building block. The numerical flux at the cell interface $x_{i+\frac{1}{2}}$ is defined as a

monotone function of two values,

${\hat{f}}_{i+\frac{1}{2}}=H\left(U_{i+\frac{1}{2}}^{-},U_{i+\frac{1}{2}}^{+}\right)$ (14)

which are referred to as the left and right boundary extrapolated values denoted

as $U_{i+\frac{1}{2}}^{-}$ and $U_{i+\frac{1}{2}}^{+}$, respectively. These extrapolated values $U_{i+\frac{1}{2}}^{-}$ and $U_{i+\frac{1}{2}}^{+}$ are approximations to $U\left(x_{i+\frac{1}{2}},t\right)$ from the left and right, respectively. They are

derived from cell averages via a high-order polynomial reconstruction that is essentially non-oscillatory.

Additionally, in order to advance the solution in time, a numerical method is employed to solve the system of ODEs (13). Typically, TVD Runge-Kutta methods are applied to maintain a consistently high-order of time accuracy. This work employs the subsequent third-order TVD Runge-Kutta method [24] :

$U_i^{\left(1\right)}=U_i^n+\text{Δ}t{L}_i\left(U_i^n\right),$ (15)

$U_i^{\left(2\right)}=\frac{3}{4}{U}_i^n+\frac{1}{4}{U}_i^{\left(1\right)}+\frac{1}{4}\text{Δ}t{L}_i\left(U_i^{\left(1\right)}\right),$ (16)

$U_i^{n+1}=\frac{1}{3}{U}_i^n+\frac{2}{3}{U}_i^{\left(2\right)}+\frac{2}{3}\text{Δ}t{L}_i\left(U_i^{\left(2\right)}\right),$ (17)

where $U_i^n$ is an approximation to the cell average ${\bar{U}}_i\left(t^n\right)$ in (12) at time $t^n$. The evaluation of the cell averages $U_i^0$ should be conducted using the exact integration of the suitable ICs, that is

$U_i^0={\bar{U}}_i\left(0\right)=\frac{1}{\text{Δ}x}{\displaystyle {\int }_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}U\left(x,0\right)\text{d}x},$ (18)

where $U\left(x,0\right)$ are the ICs.

The subsequent condition is applied to the time step $\text{Δ}t$ for every method examined in this study

$\text{Δ}t=C_{\text{CFL}}\frac{\text{Δ}x}{S_{max}^n},$ (19)

where $C_{\text{CFL}}\in \left(0,1\right]$ denotes the Courant-Friedrichs-Lewy (CFL) coefficient, $\text{Δ}x$ signifies the spatial step and $S_{\max }^n$ represents the maximal wave speed detected across the domain at time level $t=t^n$. The computation of $S_{\max }^n$ in this study is performed using the equation

$S_{\max }^n=\underset{i}{\max }\left\{\left|S_{i+\frac{1}{2}}^L\right|,\left|S_{i+\frac{1}{2}}^R\right|\right\},$ (20)

where $S_{i+\frac{1}{2}}^L$ and $S_{i+\frac{1}{2}}^R$ denote the wave speeds of the left and right non-linear

waves, respectively, in the solution of the RP ( $U_i^n,U_{i+1}^n$), which is computed using an exact Riemann solver. The speed of the head is chosen for rarefactions, while the shock speed is chosen for shocks. Refer to [2] for further information.

The spatial domain $\left[0,L\right]$ is discretized into $M$ computing cells

$I_i=\left[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}\right]$ with a dimension of $x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}=\text{Δ}x=\frac{L}{M}$, with $i=1,\cdots ,M$.

The procedures developed for the methods examined in this paper (except the RCM) are structured as follows:

1) Initial conditions. ICs are established in accordance with (18). Using the cell averages $\left\{U_i^n\right\}$ as input, our objective is to advance the solution from its starting values $\left\{U_i^n\right\}$ at time $t^n$ to new values $\left\{U_i^{n+1}\right\}$ at time $t^{n+1}=t^n+\text{Δ}t$, with a time step of size $\text{Δ}t$. To achieve this, execute the subsequent steps.

2) Boundary Conditions (BCs). BCs were implemented through the application of the periodicity condition. Whenever necessary, we positioned $r$ fictitious cells adjacent to each boundary, with the value of $r$ varying based on the method. For the left boundary, the fictitious cells are denoted by $i=0,-1,-2,\cdots$ and for the right boundary they are denoted by $i=M+1,M+2,\cdots$. Then for $j=0,1,\cdots ,r$ apply

$U_{0-j}^n=U_{1+j}^n$ (Left BCs) (21)

$U_{M+1+j}^n=U_{M-j}^n$ (Right BCs) (22)

3) Computation of time step. The time step in each method was determined in accordance with subsection 3.2. At the outset of the computations, a choice of $C_{\text{CFL}}$ must be made, which is contingent upon the method. We employ $C_{\text{CFL}}=0.45$ for the RCM, as recommended in [1] . For the WENO TVD schemes, the Courant-Friedrichs-Lewy (CFL) coefficient is usually within the range of 0.4 to 0.6. When necessary, the $C_{\text{CFL}}$ was adjusted to a lower value for the first few steps.

4) Computation of the numerical flux. This step varies based on the method used. We will cover the process of calculating the numerical flux for each method in the subsequent sections.

5) Updating of solution and advancing to the next time level. The solution is updated to the new time level $n+1$ by employing (15)-(17). After that, proceed to step 2 using $\left\{U_i^{n+1}\right\}$ as starting values.

Details regarding the value of $r$, as well as the numerical flux computation, are provided in the subsequent sections for each method.

To have a thorough understanding of the following topics, consult [10] [11] . The fundamental concept of most FV WENO methods is as follows: based on the cell averages $\left\{{\bar{u}}_j\right\}$ of a piecewise smooth function $u\left(x\right)$, determine $k$ ( $k=1,2,\cdots$) reconstruction polynomials $p_i^{\left(k,s\right)}\left(x\right)$ over the stencils $T_s\left(i\right)=\left\{I_{i-s},\cdots ,I_{i-s+k-1}\right\}$, $s=0,1,\cdots ,k-1$, of degree at most $k-1$ at each cell $I_i$ that satisfy

$\frac{1}{\text{Δ}x}{\displaystyle {\int }_{I_j}{p}_i^{\left(k,s\right)}\left(\xi \right)\text{d}\xi }={\bar{u}}_j,j=i-s,\cdots ,i-s+k-1;s=0,1,\cdots ,k-1.$

Following this, a convex combination of the $k$ reconstruction polynomials

$p_i^{\left(k,s\right)}\left(x\right)$ is employed to derive $\left(2k-1\right)$-th order approximations to the extrapolated values $u_{i+\frac{1}{2}}^{\pm }$ according to

$u_{i+\frac{1}{2}}^{-}=\underset{s=0}{\overset{k-1}{{\displaystyle \sum }}}{\omega }_s{p}_i^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right),u_{i-\frac{1}{2}}^{+}=\underset{s=0}{\overset{k-1}{{\displaystyle \sum }}}{\tilde{\omega }}_s{p}_i^{\left(k,s\right)}\left(x_{i-\frac{1}{2}}\right)\Rightarrow u_{i+\frac{1}{2}}^{+}=\underset{s=0}{\overset{k-1}{{\displaystyle \sum }}}{\tilde{\omega }}_s{p}_{i+1}^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right),$

where the coefficients ${\omega }_s$ and ${\tilde{\omega }}_s$ are referred to as nonlinear weights, and their calculation is specified for each WENO scheme in the subsequent paragraphs.

Subsequently, insert them into a numerical flux $\hat{f}\left(u_{i+\frac{1}{2}}^{-},u_{i+\frac{1}{2}}^{+}\right)$ to generate a

scheme that is analogous to (13) in the scalar case, resulting in ( $2k-1$)-th order accuracy in smooth regions. To simplify notation, we denote the ( $2k-1$)-th order WENO scheme as WENO ( $2k-1$).

For the purpose of demonstration, when $k=3$, each cell $I_i$ provides three reconstruction quadratic polynomials, $p_i^{\left(3,0\right)}\left(x\right)$, $p_i^{\left(3,1\right)}\left(x\right)$, and $p_i^{\left(3,2\right)}\left(x\right)$, over the stencils $T_0\left(i\right)=\left\{I_i,I_{i+1},I_{i+2}\right\}$, $T_1\left(i\right)=\left\{I_{i-1},I_i,I_{i+1}\right\}$, and $T_2\left(i\right)=\left\{I_{i-2},I_{i-1},I_i\right\}$, respectively. These polynomials must adhere to the following conditions

$\frac{1}{\text{Δ}x}{\displaystyle {\int }_{I_j}{p}_i^{\left(3,s\right)}\left(\xi \right)\text{d}\xi }={\bar{u}}_j,j=i-s,\cdots ,i-s+2;s=0,1,2.$

After that, the fifth-order reconstruction is obtained by

$u_{i+\frac{1}{2}}^{-}=\underset{s=0}{\overset{2}{{\displaystyle \sum }}}{\omega }_s{p}_i^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right),u_{i+\frac{1}{2}}^{+}=\underset{s=0}{\overset{2}{{\displaystyle \sum }}}{\tilde{\omega }}_s{p}_{i+1}^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right),$

An essential part of the WENO procedure is the computation of what is commonly known as Smooth or Smoothness Indicators (SIs). Denoted as ${\beta }_s^{JS}$, these indicators are defined as [9]

${\beta }_s^{JS}=\underset{l=1}{\overset{m}{{\displaystyle \sum }}}{\displaystyle {\int }_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\text{Δ}{x}^{2l-1}}{\left(\frac{{\text{d}}^l{p}_i^{\left(k,s\right)}\left(x\right)}{\text{d}{x}^l}\right)}^2\text{d}x,s=0,1,\cdots ,k-1;m=k-1.$ (23)

They can be found explicitly in [11] for $k=2,3$, and in [25] for $k=4,5$. Apart from that, they are included in Appendix C.

The reconstruction for systems can be handled by applying the WENO reconstruction procedure to each individual component of the vector $U$. Nevertheless, spurious oscillations may result from the component-wise WENO reconstruction. To prevent this, the reconstruction is conducted using local characteristic variables. The characteristic reconstruction process involves the initial transformation of the conservative variables to characteristic variables, followed by the application of the WENO reconstruction procedure to each component of these variables. The final values are acquired through the process of transforming back to conservative variables. In this study, we consistently implement reconstruction in local characteristic variables.

1) WENO ( $2k-1$)-JS TVD Scheme ( $k=2,3,4,5$)

The WENO3-JS, WENO5-JS, WENO7-JS, and WENO9-JS schemes are obtained for $k=2,3,4,5$, respectively. The WENO schemes have linear stability for $C_{\text{CFL}}\le 1$ [9] . For solutions exhibiting discontinuities, reduced CFL coefficients are generally employed, typically within the range of $C_{\text{CFL}}=0.4\text{-}0.6$. Following [20] , we use $C_{\text{CFL}}=0.4$.

Procedure 1 WENO (2k − 1)-JS TVD Scheme, k = 2, 3, 4, 5

Step 1.1. For $i=1,\cdots ,M$ set the ICs using (18).

Step 1.2. Use the cell averages ${\left\{U_i^n\right\}}_{i=1}^{i=M}$, $n\ge 0$, as input.

Step 2. Apply the BCs as specified in (21)-(22) with $r=k$.

Step 3.1. For $i=0,\cdots ,M$ transform the vector of conserved variables $U_i^n={\left[u_{1i}^n,u_{2i}^n\right]}^{\top }$ to the vector of primitive variables $U{P}_i^n={\left[h_i^n,u_i^n\right]}^{\top }={\left[u_{1i}^n,u_{2i}^n/u_{1i}^n\right]}^{\top }$.

Step 3.2. For $i=0,\cdots ,M$ solve RP ( $U{P}_i^n,U{P}_{i+1}^n$), utilising an exact Riemann solver [1] , to compute $S_{\max }^n$ using (20) and then calculate $\text{Δ}t$ according to (19).

Step 4.1. For $i=-1,\cdots ,M+1$

Step 4.1.1. Utilise (7) and (8) to calculate $R=R\left(\tilde{u},\tilde{c}\right)$ and $L=L\left(\tilde{u},\tilde{c}\right)$, where $\tilde{u}$ and $\tilde{c}$ denote the Roe average [26] between $U_i$ and $U_{i+1}$ which are defined as

$\tilde{u}=\frac{u_i\sqrt{h_i}+u_{i+1}\sqrt{h_{i+1}}}{\sqrt{h_i}+\sqrt{h_{i+1}}},\tilde{c}=\sqrt{\frac{1}{2}\left(c_i^2+c_{i+1}^2\right)},c_i=\sqrt{g{h}_i}.$

Step 4.1.2. Transform into characteristic variables using

$V_j=L{U}_j,j=i-k+1,\cdots ,i+k.$

Step 4.1.3. To estimate the left extrapolated value $V_{i+\frac{1}{2}}^{-}$ calculate

Step 4.1.3.1. $p_i^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right)={\displaystyle {\sum }_{j=0}^{k-1}{c}_{k,s,j}{V}_{i-s+j}}$, $s=0,1,\cdots ,k-1$,

Step 4.1.3.2. ${\beta }_s^{JS}={\beta }_s^{JS}\left(V,i\right)$, $s=0,1,\cdots ,k-1$, employing (30)-(33),

contingent upon k.

Step 4.1.3.3. ${\alpha }_s^{JS}=\frac{d_{k,s}}{{\left(ϵ+{\beta }_s^{JS}\right)}^2}$, $s=0,1,\cdots ,k-1$,

Step 4.1.3.4. ${\omega }_s^{JS}=\frac{{\alpha }_s^{JS}}{{\displaystyle {\sum }_{j=0}^{k-1}{\alpha }_j^{JS}}}$, $s=0,1,\cdots ,k-1$, (nonlinear weights).

Step 4.1.3.5. Finally, $V_{i+\frac{1}{2}}^{-}={\displaystyle {\sum }_{s=0}^{k-1}{\omega }_s^{JS}{p}_i^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

The constant $ϵ$ is introduced to prevent the denominator from becoming zero. The coefficients $c_{k,s,j}$ are provided in [11] , whereas the values of $d_{k,s}$ can be located in [11] for $k=2,3$, and in [25] for $k=4,5$. Table A1 and Table A2 of Appendix B contain them, as well. In the literature, the coefficients $d_{k,s}$ are commonly referred to as ideal, optimal, or linear weights. For WENO-JS TVD scheme we take $ϵ={10}^{-20}$ [20] .

Step 4.1.4. To estimate the right extrapolated value $V_{i+\frac{1}{2}}^{+}$ calculate

Step 4.1.4.1. $p_{i+1}^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right)={\displaystyle {\sum }_{j=0}^{k-1}{c}_{k,s-1,j}{V}_{i+1-s+j}}$, $s=0,1,\cdots ,k-1$,

Step 4.1.4.2. ${\beta }_s^{JS}={\beta }_s^{JS}\left(V,i+1\right)$, $s=0,1,\cdots ,k-1$, employing (30),

Step 4.1.4.3. ${\alpha }_s^{JS}=\frac{d_{k-1-s}}{{\left(ϵ+{\beta }_s^{JS}\right)}^2}$, $s=0,1,\cdots ,k-1$,

Step 4.1.4.4. ${\omega }_s^{JS}=\frac{{\alpha }_s^{JS}}{{\displaystyle {\sum }_{j=0}^{k-1}{\alpha }_j^{JS}}}$, $s=0,1,\cdots ,k-1$.

Step 4.1.4.5. Thus, $V_{i+\frac{1}{2}}^{+}={\displaystyle {\sum }_{s=0}^{k-1}{\omega }_s^{JS}{p}_{i+1}^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Step 4.1.5. Transform back into conservative variables

$U_{i+\frac{1}{2}}^{\pm }=R{V}_{i+\frac{1}{2}}^{\pm }={\left[u_{1i+\frac{1}{2}}^{\pm },u_{2i+\frac{1}{2}}^{\pm }\right]}^{\top }.$

Step 4.1.6. Convert to primitive variables

$U{P}_{i+\frac{1}{2}}^{\pm }={\left[u_{1i+\frac{1}{2}}^{\pm },u_{2i+\frac{1}{2}}^{\pm }/u_{1i+\frac{1}{2}}^{\pm }\right]}^{\top }={\left[h_{i+\frac{1}{2}}^{\pm },u_{i+\frac{1}{2}}^{\pm }\right]}^{\top }.$

Step 4.1.7. Solve RP ( $U{P}_{i+\frac{1}{2}}^{-},U{P}_{i+\frac{1}{2}}^{+}$) utilising an HLL Riemann [1] [27] solver and store the following values

$c_{1,i+\frac{1}{2}}=S_{i+\frac{1}{2}}^L\frac{\text{Δ}t}{\text{Δ}x},c_{2,i+\frac{1}{2}}=S_{i+\frac{1}{2}}^R\frac{\text{Δ}t}{\text{Δ}x},\text{Δ}{h}_{1,i+\frac{1}{2}}={\hat{h}}^{\text{*}}-h_{i+\frac{1}{2}}^{-},$ (24)

$\text{Δ}{h}_{2,i+\frac{1}{2}}=h_{i+\frac{1}{2}}^{+}-{\hat{h}}^{\text{*}},S{F}_{i+\frac{1}{2}}=F_{i+\frac{1}{2}}^{-}+F_{i+\frac{1}{2}}^{+},$ (25)

$\text{Δ}{F}_{2,i+\frac{1}{2}}=F_{i+\frac{1}{2}}^{+}-{\hat{F}}^{\text{*}},\text{Δ}{F}_{1,i+\frac{1}{2}}={\hat{F}}^{\text{*}}-F_{i+\frac{1}{2}}^{-},$ (26)

where $F_{i+\frac{1}{2}}^{-}=F\left(U_{i+\frac{1}{2}}^{-}\right)$, $F_{i+\frac{1}{2}}^{+}=F\left(U_{i+\frac{1}{2}}^{+}\right)$. The values ${\hat{h}}^{\text{*}}$ and ${\hat{F}}^{\text{*}}$ are estimates

of the exact solutions for $h^{\text{*}}$ and $F\left(U^{\text{*}}\right)$ in the star region, respectively. The two-rarefaction Riemann solver [1] is employed to calculate ${\hat{h}}^{\text{*}}$.

Step 4.2. For $i=0,\cdots ,M$ calculate

4.2.1. $r_{j,i+\frac{1}{2}}=\{\begin{array}{l}\frac{\text{Δ}{h}_{j,i-\frac{1}{2}}}{\text{Δ}{h}_{j,i+\frac{1}{2}}}\text{if}{c}_{j,i+\frac{1}{2}}>0\hfill \\ \frac{\text{Δ}{h}_{j,i+\frac{3}{2}}}{\text{Δ}{h}_{j,i+\frac{1}{2}}}\text{if}{c}_{j,i+\frac{1}{2}}<0\hfill \end{array},j=1,2$ (27)

and store the subsequent values

4.2.2. ${\hat{f}}_{i+\frac{1}{2}}=\frac{1}{2}S{F}_{i+\frac{1}{2}}-\frac{1}{2}\underset{j=1}{\overset{2}{{\displaystyle \sum }}}\text{sign}\left(c_{j,i+\frac{1}{2}}\right){\varphi }_{i+\frac{1}{2}}\left(r_{j,i+\frac{1}{2}},c_{j,i+\frac{1}{2}}\right)\text{Δ}{F}_{j,i+\frac{1}{2}}$, (28)

where $\varphi =\varphi \left(r,c\right)$ is a WAF limiter function that corresponds to the well-known flux limiter SUPERBEE of Roe [28] and is defined as

$\varphi \left(r,c\right)=\{\begin{array}{ll}1\hfill & \text{if}r\le 0,\hfill \\ 1-2\left(1-\left|c\right|\right)r\hfill & \text{if}0<r<\frac{1}{2},\hfill \\ \left|c\right|\hfill & \text{if}\frac{1}{2}\le r\le 1,\hfill \\ 1-\left(1-\left|c\right|\right)r\hfill & \text{if}1<r<2,\hfill \\ 2\left|c\right|-1\hfill & \text{if}r\ge 2.\hfill \end{array}$ (29)

Step 4.3. For $i=1,\cdots ,M$ store the values $L_i\left(U_i^n\right)=-\frac{1}{\text{Δ}x}\left({\hat{f}}_{i+\frac{1}{2}}-{\hat{f}}_{i-\frac{1}{2}}\right)$.

Step 5.1. For $i=1,\cdots ,M$ apply (15) to compute $U_i^{\left(1\right)}$.

Step 5.2. Retain the values of $\left\{U_i^n\right\}$ and $\left\{U_i^{\left(1\right)}\right\}$, then execute Steps 1.2-4.3 by setting $\left\{U_i^{\left(1\right)}\right\}$ as initial values. Once $L_i\left(U_i^{\left(1\right)}\right)$ have been calculated, utilise (16), for $i=1,\cdots ,M$, to produce $U_i^{\left(2\right)}$.

Step 5.3. Preserve the values of $\left\{U_i^{\left(2\right)}\right\}$, and imitate Steps 1.2-4.3 by assigning $\left\{U_i^{\left(2\right)}\right\}$ as the starting values. After $L_i\left(U_i^{\left(2\right)}\right)$ have been computed, apply (17), for $i=1,\cdots ,M$, to obtain $U_i^{n+1}$.

Step 5.4. Utilising $\left\{U_i^{n+1}\right\}$ as initial values, proceed to Step 1.2 to advance to the next time level.

2) WENO5-M TVD Scheme

At certain smooth extrema or near critical points, the smoothness indicators-based nonlinear weights ${\omega }_s^{JS}$, $k=3$, are unable to meet the sufficient condition for fifth-order accuracy. Henrick et al. [12] introduced mapping functions to address this issue by applying them to the nonlinear weights ${\omega }_s^{JS}$. In addition to meeting the fifth-order accuracy requirement with the resulting nonlinear weights ${\omega }_s^M$, the mapped WENO scheme yields more accurate numerical solutions around discontinuities. Higher-order WENO-M schemes were also devised in [29] . For the WENO5-M TVD scheme, we use $ϵ={10}^{-40}$ [12] and $C_{\text{CFL}}=0.4$ [20] .

Procedure 2 WENO5-M TVD Scheme

Steps 1.1-4.1.3.4. Execute Steps 1.1-4.1.3.4 of Procedure 1 for $k=3$.

Step 4.1.3.5. ${\alpha }_s^M=g_s\left({\omega }_s^{JS}\right)=\frac{{\omega }_s^{JS}\left(d_{3,s}+d_{3,s}^2-3d_{3,s}{\omega }_s^{JS}+{\left({\omega }_s^{JS}\right)}^2\right)}{d_{3,s}^2+{\omega }_s^{JS}\left(1-2d_{3,s}\right)}$, $s=0,1,2$.

Step 4.1.3.6. ${\omega }_s^M=\frac{{\alpha }_s^M}{{\displaystyle {\sum }_{j=0}^2{\alpha }_j^M}}$, $s=0,1,2$, (mapped weights).

Step 4.1.3.7. $V_{i+\frac{1}{2}}^{-}={\displaystyle {\sum }_{s=0}^2{\omega }_s^M{p}_i^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.4-4.1.4.4. Execute Steps 4.1.4-4.1.4.4 of Procedure 1 for $k=3$.

Steps 4.1.4.5-6. Compute ${\alpha }_s^M$ and ${\omega }_s^M$ in the same manner as in Steps 4.1.3.5-6.

Step 4.1.4.7. $V_{i+\frac{1}{2}}^{+}={\displaystyle {\sum }_{s=0}^2{\omega }_s^M{p}_{i+1}^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.5-5.4. Execute Steps 4.1.5-5.4 of Procedure 1.

3) WENO5-Z TVD Scheme

Borges et al. developed a different technique in [13] that enabled them to achieve fifth-order accuracy at the critical points of smooth solutions without the need of a map. Their introduction of the global SI and the development of Z-type nonlinear weights, ${\omega }_s^Z$, led to the WENO-Z scheme. Compared to WENO5-JS and WENO5-M, the WENO5-Z scheme that results is less dissipative. The development of higher-order WENO-Z schemes was presented in [30] . We employ $ϵ={10}^{-40}$ and $C_{\text{CFL}}=0.4$ [13] for the WENO5-Z TVD scheme.

Procedure 3 WENO5-Z TVD Scheme

Steps 1.1-4.1.3.2. Implement Steps 1.1-4.1.3.2 of Procedure 1 for $k=3$.

Step 4.1.3.3. ${\tau }_5=\left|{\beta }_0^{JS}-{\beta }_2^{JS}\right|$ (global SI).

Step 4.1.3.4. ${\beta }_s^Z=\frac{{\beta }_s^{JS}+ϵ}{{\beta }_s^{JS}+{\tau }_5+ϵ}$, $s=0,1,2$.

Step 4.1.3.5. ${\alpha }_s^Z=\frac{d_{3,s}}{{\beta }_s^Z}=d_{3,s}\left(1+\frac{{\tau }_5}{{\beta }_s^{JS}+ϵ}\right)$, $s=0,1,2$.

Step 4.1.3.6. ${\omega }_s^Z=\frac{{\alpha }_s^Z}{{\displaystyle {\sum }_{j=0}^2{\alpha }_j^Z}}$, $s=0,1,2$.

Step 4.1.3.7. $V_{i+\frac{1}{2}}^{-}={\displaystyle {\sum }_{s=0}^2{\omega }_s^Z{p}_i^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.4-4.1.4.2. Implement Steps 4.1.4-4.1.4.2 of Procedure 1 for $k=3$.

Steps 4.1.4.3-6. Calculate ${\tau }_5$, ${\beta }_s^Z$, ${\alpha }_s^Z$ and ${\omega }_s^Z$, $s=0,1,2$, in the same manner as described in Steps 4.1.3.3-6 above.

Step 4.1.4.7. $V_{i+\frac{1}{2}}^{+}={\displaystyle {\sum }_{s=0}^2{\omega }_s^Z{p}_{i+1}^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.5-5.4 Implement Steps 4.1.5-5.4 of Procedure 1.

4) WENO5-NS TVD Scheme

The WENO5-M and WENO5-Z schemes are developed by utilising the well-known SIs of WENO5-JS, ${\beta }_s^{JS}$, in an innovative manner. In contrast to the SIs ${\beta }_s^{JS}$, which are constructed via the $L_2$-norm, Ha et al. introduce an alternative version of SIs that is based on the $L_1$-norm in [14] . Their explicit expression is located in [14] and has been provided as well in Appendix D. The WENO5-NS TVD scheme is implemented with $ϵ={10}^{-40}$ and $C_{\text{CFL}}=0.5$ [14] .

Procedure 4 WENO5-NS TVD Scheme

Steps 1.1-4.1.3.1. Perform Steps 1.1-4.1.3.1 of Procedure 1 for $k=3$.

Step 4.1.3.2. ${\beta }_s^{NS}={\beta }_s^{NS}\left(V,i\right)$, $s=0,1,2$, employing (34).

Step 4.1.3.3. $\zeta =\zeta \left(i\right)=\frac{1}{2}\left({\left|{\beta }_0^{NS}-{\beta }_2^{NS}\right|}^2+g{\left(\left|V_{i+1}-V_i\right|\right)}^2\right)$, $g\left(x\right)=\frac{x^3}{x^3+1}$.

Step 4.1.3.4. ${\alpha }_s^{NS}=d_{3,s}\left(1+\frac{\zeta }{ϵ+{\beta }_s^{NS}}\right)$, $s=0,1,2$.

Step 4.1.3.5. ${\omega }_s^{NS}=\frac{{\alpha }_s^{NS}}{{\displaystyle {\sum }_{j=0}^2{\alpha }_j^{NS}}}$, $s=0,1,2$.

Step 4.1.3.6. $V_{i+\frac{1}{2}}^{-}={\displaystyle {\sum }_{s=0}^2{\omega }_s^{NS}{p}_i^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.4-4.1.4.1. Perform Steps 4.1.4-4.1.4.1 of Procedure 1.

Steps 4.1.4.2-5. Compute ${\beta }_s^{NS}\left(V,i+1\right)$, $\zeta \left(i+1\right)$, ${\alpha }_s^{NS}$, and ${\omega }_s^{NS}$, $s=0,1,2$, using the formulas given in Steps 4.1.3.2-5.

Step 4.1.4.6. $V_{i+\frac{1}{2}}^{+}={\displaystyle {\sum }_{s=0}^2{\omega }_s^{NS}{p}_{i+1}^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.5-5.4 Perform Steps 4.1.5-5.4 of Procedure 1.

5) WENO5-P TVD Scheme

Kim et al. introduce a novel fifth-order WENO in [15] that enhances the WENO5-NS in [14] . They achieve this by simplifying WENO5-NS’s nonlinear weights and modifying its SIs. We utilise the values $ϵ={10}^{-40}$ [15] and $C_{\text{CFL}}=0.4$ [20] for the WENO5-P TVD scheme.

Procedure 5 WENO5-P TVD Scheme

Steps 1.1-4.1.3.2. Run Steps 1.1-4.1.3.2 of Procedure 4.

Step 4.1.3.3. ${\beta }_0^P={\beta }_0^{NS}$, ${\beta }_1^P=\left(1+\delta \right){\beta }_1^{NS}$, ${\beta }_2^P=\left(1-\delta \right){\beta }_2^{NS}$, $\delta =0.05$ as per [15] .

Step 4.1.3.4. $\zeta ={\left({\beta }_0^{NS}-{\beta }_2^{NS}\right)}^2$.

Step 4.1.3.5. ${\alpha }_s^P=d_{3,s}\left(1+\frac{\zeta }{{\left(ϵ+{\beta }_s^P\right)}^2}\right)$, $s=0,1,2$.

Step 4.1.3.6. ${\omega }_s^P=\frac{{\alpha }_s^P}{{\displaystyle {\sum }_{j=0}^2{\alpha }_j^P}}$, $s=0,1,2$.

Step 4.1.3.7. $V_{i+\frac{1}{2}}^{-}={\displaystyle {\sum }_{s=0}^2{\omega }_s^P{p}_i^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.4-4.1.4.2. Run Steps 4.1.4-4.1.4.2 of Procedure 4.

Steps 4.1.4.3-6. Evaluate, ${\beta }_s^P$, $\zeta$, ${\alpha }_s^P$, and ${\omega }_s^P$, $s=0,1,2$, in the same manner as described in Steps 4.1.3.3-6 above.

Step 4.1.4.7. $V_{i+\frac{1}{2}}^{+}={\displaystyle {\sum }_{s=0}^2{\omega }_s^P{p}_{i+1}^{\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)}$.

Steps 4.1.5-5.4 Run Steps 4.1.5-5.4 of Procedure 1.

6) WENO5-ZQ TVD Scheme

The WENO5-ZQ TVD scheme employs different reconstruction polynomials, designated $p_i^{ZQ\left(3,s\right)}\left(x\right)$, as opposed to the conventional ones, $p_i^{\left(3,s\right)}\left(x\right)$, $s=0,1,2$. In fact, $p_i^{ZQ\left(3,0\right)}\left(x\right)=p_i^{\left(5,2\right)}\left(x\right)$, $p_i^{ZQ\left(3,1\right)}\left(x\right)=p_i^{\left(2,1\right)}\left(x\right)$, and $p_i^{ZQ\left(3,2\right)}\left(x\right)=p_i^{\left(2,0\right)}\left(x\right)$. To determine the SIs, the authors in [16] utilised Equation (23), with $p_i^{\left(k,s\right)}\left(x\right)=p_i^{ZQ\left(3,s\right)}\left(x\right)$ and the values of $m$ being 4 for $s=0$, and 1 for $s=2,3$. As a result, if we represent the SIs of WENO $\left(2k-1\right)$-JS as ${\beta }_{k,s}^{JS}$, we can see that the SIs of WENO5-ZQ, ${\beta }_s^{ZQ}$, are: ${\beta }_0^{ZQ}={\beta }_{5,2}^{JS}$, ${\beta }_1^{ZQ}={\beta }_{2,1}^{JS}$ and ${\beta }_2^{ZQ}={\beta }_{2,0}^{JS}$. For the WENO5-ZQ TVD scheme, we use $ϵ={10}^{-6}$ and $C_{\text{CFL}}=0.6$ [16] .

Procedure 6 WENO5-ZQ TVD Scheme

Steps 1.1-4.1.3 Execute Steps 1.1-4.1.3 of Procedure 1 for $k=3$.

Step 4.1.3.1. $p_i^{ZQ\left(3,0\right)}\left(x_{i+\frac{1}{2}}\right)=p_i^{\left(5,2\right)}\left(x_{i+\frac{1}{2}}\right)$, $p_i^{ZQ\left(3,1\right)}\left(x_{i+\frac{1}{2}}\right)=p_i^{\left(2,1\right)}\left(x_{i+\frac{1}{2}}\right)$, $p_i^{ZQ\left(3,2\right)}\left(x_{i+\frac{1}{2}}\right)=p_i^{\left(2,0\right)}\left(x_{i+\frac{1}{2}}\right)$.

The computation of $p_i^{\left(k,s\right)}\left(x_{i+\frac{1}{2}}\right)$ can be derived from Step 4.1.3.1 of Procedure 1.

Step 4.1.3.2. ${\beta }_0^{ZQ}={\beta }_{5,2}^{JS}\left(V,i\right)$, ${\beta }_1^{ZQ}={\beta }_{2,1}^{JS}\left(V,i\right)$, ${\beta }_2^{ZQ}={\beta }_{2,0}^{JS}\left(V,i\right)$, from (33), (30).

Step 4.1.3.3. $\tau ={\left(\frac{\left|{\beta }_1^{ZQ}-{\beta }_2^{ZQ}\right|+\left|{\beta }_1^{ZQ}-{\beta }_3^{ZQ}\right|}{2}\right)}^2$.

Step 4.1.3.4. ${\alpha }_s^{ZQ}={\gamma }_s\left(1+\frac{\tau }{ϵ+{\beta }_s^{ZQ}}\right)$, $s=0,1,2$, where ${\gamma }_0=0.98$,

${\gamma }_1={\gamma }_2=0.01$, as indicated by [16] .

Step 4.1.3.5. ${\omega }_s^{ZQ}=\frac{{\alpha }_s^{ZQ}}{{\displaystyle {\sum }_{j=0}^2{\alpha }_j^{ZQ}}}$, $s=0,1,2$.

Step 4.1.3.6. $\begin{array}{c}{V}_{i+\frac{1}{2}}^{-}={\omega }_0^{ZQ}(\frac{1}{{\gamma }_0}{p}_i^{ZQ\left(3,0\right)}\left(x_{i+\frac{1}{2}}\right)-\frac{{\gamma }_1}{{\gamma }_0}{p}_i^{ZQ\left(3,1\right)}\left(x_{i+\frac{1}{2}}\right)\\ -\frac{{\gamma }_2}{{\gamma }_0}{p}_i^{ZQ\left(3,2\right)}\left(x_{i+\frac{1}{2}}\right))+\underset{s=1}{\overset{2}{{\displaystyle \sum }}}{\omega }_s^{ZQ}{p}_i^{ZQ\left(3,s\right)}\left(x_{i+\frac{1}{2}}\right)\end{array}$.