Iterative Solver of the Wet-Bed Step Riemann Problem

Abstract

This study presents a one-dimensional solver of the shallow water equations designed for the wet-bed step Riemann problem. Nonlinear mass and momentum equations incorporating shock and rarefaction waves in a straight one-dimensional channel are expressed as a pair of equations that depend solely on local depth values either side of the step. These unified equations are uniquely designed for the four conditions involving shock and rarefaction waves that can occur in the Step Riemann Problem. The Levenberg–Marquardt method is used to solve these simplified nonlinear equations. Four verification tests are considered for shallow free surface flow in a wet-bed channel with a step. These cases involve two rarefactions, opposing shock-like hydraulic bores, and a rarefaction and shock-like bore. The numerical predictions are in close agreement with existing theory, demonstrating that the method is very effective at solving the wet-bed step Riemann problem.

1. Introduction

The shallow water equations (SWEs), obtained by depth-integration of the continuity and Navier–Stokes (NS) equations [1,2,3], are extensively utilized in the simulation of river flows. Many different solvers have been proposed for the shallow water equations. For example, Bender and Öffner [4] developed entropy-conservative Discontinuous Galerkin (DG) methods for solving the stochastic hyperbolic shallow water equations, which propagate uncertainty from input to output using generalized polynomial chaos. In the context of machine learning, Yao et al. [5] proposed a hybrid theory-data-driven modeling technique based on the SWEs which combined physical laws with data-driven corrections to enhance model accuracy and adaptability. Yao et al. demonstrated the effectiveness of their deep learning method for an idealized dam break flow over a flat bed, and simulated an actual landslide that occurred at Yichang, China in 2019.

Both the SWEs and their inviscid counterparts, the Euler equations, belong to nonlinear hyperbolic systems of conservation laws for which advanced solvers have been developed for problems involving steep or even discontinuous gradients in the dependent variables (e.g., flow depth and mass flux). For example, Li et al. [6] proposed a high-order hybrid Weighted Essentially Non-Oscillatory (WENO) scheme, termed WENO-MS, that combines third- and fifth-order finite-difference modified WENO schemes for high-order reconstructions in different flow regions, enabling the WENO scheme to capture shocks while properly modeling smooth flow regions. This approach offers improved robustness and efficiency compared to classical WENO schemes. Zhang et al. [7] developed a finite difference scheme that maintains the entropy condition for hyperbolic systems, enhancing the stability and accuracy of their simulations both at discontinuities and in smooth flow regions. Zahran and Abdalla [8] proposed a central random choice method (CRCM) that merges central Arbitrary DERivatives (ADER) techniques with random sampling, resulting in a scheme that is more accurate, simpler to implement, and requires less computational memory than earlier conventional schemes. Zehran and Abdalla verified their scheme for a wide range of benchmark tests including vortex evolution, double Mach reflection, and shock flow past a forward-facing step. Given that traditional polynomial-based schemes may face challenges in capturing complex flow features, Fu et al. [9] proposed the non-polynomial-based Targeted Essentially Non-Oscillatory (TENO) scheme to address such issues, offering improved resolution with respect to small-scale structures and discontinuities in compressible flows such as occur in double Mach reflection, etc. [10]. Deng et al. [11] recently developed a high-resolution shock-capturing scheme called Reconstruction Operators on Unified-Normalise-variable Diagram (ROUND) for application on non-uniform meshes. ROUND overcomes the previous requirement to calculate smoothness indicators, and is proven to be accurate, efficient and robust when applied to high-speed compressible flows involving both small-scale flow structures and strong shock waves.

To further enhance the numerical efficiency of nonlinear hyperbolic solvers, Leveque proposed the Large Time Step (LTS) scheme which overcomes the limitation of the Courant–Friedrichs–Lewy (CFL) condition that CFL < 1 for numerical solvers of scalar hyperbolic conservation laws [12]. The scheme was expanded to the SWEs by Murillo et al. [13] and Morales-Hernández and Murillo [14] using a rarefaction splitting method based on an Approximate Riemann Solver (ARS) to avoid entropy violation. The scheme was later extended to 2D on both structured [15] and unstructured grids [16]. Xu et al. [17] introduced an Exact Riemann Solver (ERS) to LTS which better split the rarefaction waves and enabled a larger CFL number to be attained, thus enhancing computational efficiency. Qian et al. [18] extended LTS to the Euler equations by including nonlinear wave interactions and a multi-wave approximation for rarefaction processes. An LTS wave adding scheme was proposed by Dong and Liu [19] who later developed a second-order LTS version [20]. Lindqvist et al. [21] identified the least and most diffusive total variation diminishing (TVD) schemes in an LTS framework and proposed a one-parameter family of LTS-TVD schemes. LTS was also applied to the simulation of a two-layer fluid problem by Prebeg et al. [22]. Harten [23] suggested an alternative LTS scheme which coalesced multiple steps into a single step by operator merging. Harten’s approach was extended to the SWEs by Xu et al. [24] and the Euler equations by Qian and Lee [25].

Unlike the Euler equations, which are also nonlinear hyperbolic systems derived from continuity and the Navier–Stokes equations, the SWEs are inhomogeneous. In engineering practice, source terms, related to bed topography and bed resistance, have to be evaluated using specially designed methods when simulating actual shallow flows. For example, García-Navarro and Cendón [26] retained only part of the split flux difference depending on the conserved variables when determining the convective flux. Rogers et al. [27] presented an algebraic technique that balanced the flux gradient and source terms when using a Roe scheme. Liang and Borthwick [28] proposed a stage-discharge method with flux and source terms that were properly balanced when computing shallow flows with wet-dry fronts over complex topography. Zhang et al. [29] developed well-balanced, energy-stable, adaptive moving mesh finite difference schemes that maintain the ‘lake at rest’ steady state and are adaptable to shallow flow over varying bed topography.

Another method that can handle the riverbed source term involves a direct solver of the Step Riemann Problem (SRP). Although great effort has been put into developing a high-precision Riemann solver of the SRP, the choice of solver remains a subject of debate, with two approaches commonly used. One method, suggested by Alcrudo and Benkhaldoun [30] and Bukreev et al. [31], utilizes conservation of mass and energy principles, and assumes that the riverbed slope tends to infinity at the discontinuity in the SRP, invalidating the momentum conservation equation. In practice, turbulence causes energy dissipation, meaning that energy is not conserved during flow over the step, leading to a flaw in the foregoing theory. Galloüet et al. [32], Chinnayya et al. [33], and Andrianov [34] noted that the Riemann invariant is conserved in characteristic space because the contact discontinuity is a standing wave, thus inevitably enforcing conservation of mass and energy. Aleksyuk and Belikov [35,36] addressed the issue of multiple solutions arising from conservation of mass and energy by introducing a condition for the conservation of mass at a discontinuity. The second approach employs conservation of mass and momentum. For example, Bernetti et al. [37] imposed energy conservation as a constraint to verify the validity of numerical predictions obtained using mass-momentum conservation. Xu et al. [38] obtained a high-precision solution for mass and momentum conservation by applying Levenberg–Marquardt (LM) iteration. Using mass and momentum conservation, Rosatti and Begnudelli [39] discovered that when the upstream riverbed is at a higher elevation than the downstream riverbed, the flow energy increases rather than decreases, which contradicts the laws of physics. As a result, this approach remains contentious, and no definitive conclusion has been reached.

The present study applies the second of the two SRP approaches discussed above to shallow flow over a step in a one-dimensional channel with a wet-bed throughout. Our paper advances the subject through the derivation of unified equations specifically designed for the four conditions involving shock and rarefaction waves that can occur in the Step Riemann Problem. The structure of our paper is as follows. Section 2 introduces shock and rarefaction conditions and describes their integration. Section 3 derives simplified equations that apply to the four Riemann states for the SRP. Section 4 describes the results obtained for four benchmark test problems. Finally, Section 5 summarizes the main findings of this study.

2. Step Riemann Problem

We consider shallow flow in one spatial dimension along a straight open channel. Assuming hydrostatic pressure and ignoring friction, the shallow water equations (Saint-Venant equations) [40] derived by depth-averaging the continuity and NS equations can be expressed as follows:

$${\displaystyle \frac{\partial U}{\partial t}}+{\displaystyle \frac{\partial F\left(U\right)}{\partial x}}=S_f$$

(1)

where

$$U=\left[\begin{array}{c}h\\ hu\end{array}\right],F\left(U\right)=\left[\begin{array}{c}hu\\ h{u}^2+{\displaystyle \frac{1}{2}}g{h}^2\end{array}\right],S_f=\left[\begin{array}{c}0\\ gh{\displaystyle \frac{\partial z}{\partial x}}\end{array}\right]$$

in which U is the vector of dependent variables, F(U) is the vector of horizontal mass and momentum fluxes, t is time, x is horizontal distance along the channel, z is bed elevation, h is the water depth, u is the average flow velocity, and g is gravitational acceleration.

Figure 1 provides a detailed illustration of the SRP for a sudden step of height ∆z in the channel bed. Here, the bed is wet throughout the channel. Local depth, bulk velocity, and bed elevation are represented by (h_L, U_L, and z_L) to the immediate left (upstream) of the step and (h_R, U_R, and z_R) to the immediate right (downstream) of the step.

Figure 2 illustrates one of four possible Riemann solutions for the SRP. At the step, a static shock occurs with zero celerity. Immediately either side of the static shock, the variables u_SL, h_SL, u_SR, and h_SR are unknown and can be determined from relations with known variables u_L, h_L, z_L, u_R, and h_R.

If the left side is a rarefaction wave, the Riemann invariants are as follows:

$$u_L+2\sqrt{g{h}_L}=u_S+2\sqrt{g{h}_S}.$$

(2)

If the left side is a shock-like hydraulic bore, the Riemann invariants are as follows:

$$\left\{\begin{array}{l}{h}_L{u}_L-h_{SL}{u}_{SL}=s\left(h_L-h_{SL}\right)\\ \left(h_L{u}_L^2+{\displaystyle \frac{1}{2}}g{h}_L^2\right)-\left(h_{SL}{u}_{SL}^2+{\displaystyle \frac{1}{2}}g{h}_{SL}^2\right)=s\left(h_L{u}_L-h_{SL}{u}_{SL}\right)\end{array}\right.$$

(3)

where s is the speed of the shock.

If the right side is a rarefaction wave, then

$$u_R-2\sqrt{g{h}_R}=u_S-2\sqrt{g{h}_S}.$$

(4)

If the right side is a shock, then

$$\left\{\begin{array}{l}{h}_R{u}_R-h_{SR}{u}_{SR}=s\left(h_R-h_{SR}\right)\\ \left(h_R{u}_R^2+{\displaystyle \frac{1}{2}}g{h}_R^2\right)-\left(h_{SR}{u}_{SR}^2+{\displaystyle \frac{1}{2}}g{h}_{SR}^2\right)=s\left(h_R{u}_R-h_{SR}{u}_{SR}\right)\end{array}\right..$$

(5)

The static shock satisfies the following relationships:

$$h_{SL}{u}_{SL}=h_{SR}{u}_{SR}$$

(6)

$$h_{SL}{u}_{SL}^2+{\displaystyle \frac{1}{2}}g{h}_{SL}^2=h_{SR}{u}_{SR}^2+{\displaystyle \frac{1}{2}}g{h}_{SR}^2+D$$

(7)

where

$$D=g\left(h_k-{\displaystyle \frac{\left|Z_R-Z_L\right|}{2}}\right)\left(Z_R-Z_L\right),k=\left\{\begin{array}{ll}L& Z_L\le Z_R\\ R& Z_L\ge Z_R\end{array}\right..$$

(8)

Table 1. Conventional equations of Riemann solver for wet-bed SRP.

	Right-Interface-Left	Abbr.	Equations
1	Rarefaction-static shock-shock	RBS	(2) (6) (7) (5)
2	Shock-static shock-shock	SBS	(3) (6) (7) (5)
3	Rarefaction-static shock-Rarefaction	RBR	(2) (6) (7) (4)
4	Shock-static shock-Rarefaction	SBR	(3) (6) (7) (4)

summarizes the Riemann states for the SRP.

3. Solution Methodology

We rewrite Equation (2) as

$$u_{SL}=D_1-2\sqrt{g{h}_{SL}},$$

(9)

where

$$D_1=u_L+2\sqrt{g{h}_L}.$$

(10)

Taking the derivative of Equation (9) with respect to h_SL, then

$${\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}=-\sqrt{{\displaystyle \frac{g}{h_{SL}}}}.$$

(11)

Similarly, Equation (4) is rewritten as

$$u_{SR}=D_2+2\sqrt{g{h}_{SR}},$$

(12)

in which

$$D_2=u_R+2\sqrt{g{h}_R}.$$

(13)

Differentiating Equation (12) with respect to h_SR yields the following:

$${\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}=\sqrt{{\displaystyle \frac{g}{h_{SR}}}}.$$

(14)

Setting

$$\left\{\begin{array}{c}{\widehat{u}}_L=u_L-s_L\\ {\widehat{u}}_{\mathit{SL}}=u_{SL}-s_L\end{array}\right.,$$

(15)

Equation (3) becomes:

$$h_L{\widehat{u}}_L=h_{SL}{\widehat{u}}_{SL}$$

(16)

and

$$h_L{\widehat{u}}_L^2+{\displaystyle \frac{1}{2}}g{h}_L^2=h_{SL}{\widehat{u}}_{SL}^2+{\displaystyle \frac{1}{2}}g{h}_{SL}^2.$$

(17)

Combining Equations (16) and (17):

$$\left(h_L{\widehat{u}}_L\right){\widehat{u}}_L-\left(h_{SL}{\widehat{u}}_{SL}\right){\widehat{u}}_{SL}={\displaystyle \frac{1}{2}}g{h}_{SL}^2-{\displaystyle \frac{1}{2}}g{h}_L^2$$

(18)

where

$$M_L=h_L{\widehat{u}}_L=h_{SL}{\widehat{u}}_{SL}.$$

(19)

Substituting Equations (16) and (19) into Equation (18) gives

$$M_L\left({\widehat{u}}_L-{\widehat{u}}_{SL}\right)={\displaystyle \frac{1}{2}}g{h}_{SL}^2-{\displaystyle \frac{1}{2}}g{h}_L^2.$$

(20)

Rearranging,

$$M_L=-{\displaystyle \frac{1}{2}}g{\displaystyle \frac{h_{SL}^2-h_L^2}{{\widehat{u}}_L-{\widehat{u}}_{SL}}}.$$

(21)

From Equation (19):

$${\widehat{u}}_L={\displaystyle \frac{M_L}{h_L}}, {\widehat{u}}_{SL}={\displaystyle \frac{M_L}{h_{SL}}}.$$

(22)

Substituting Equation (22) into Equation (21) yields:

$$M_L=\sqrt{{\displaystyle \frac{1}{2}}g\left(h_{SL}+h_L\right)h_{SL}{h}_L}.$$

(23)

Based on Equation (15)

$${\widehat{u}}_{SL}-{\widehat{u}}_L=u_{SL}-u_L.$$

(24)

Substituting Equation (24) into Equation (21) gives:

$$u_{SL}=u_L-{\displaystyle \frac{1}{2}}g{\displaystyle \frac{h_{SL}^2-h_L^2}{M_L}}.$$

(25)

Substituting Equation (23) into Equation (25) gives:

$$u_{SL}=u_L-\left(h_{SL}-h_L\right)D_3.$$

(26)

Differentiating Equations (9) and (26) with respect to h_SL yields:

$${\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}=-D_3+g{\displaystyle \frac{\left(h_{SL}-h_L\right)}{4D_3{h}_{SL}^2}}$$

(27)

where

$$D_3=\sqrt{{\displaystyle \frac{1}{2}}g{\displaystyle \frac{h_{SL}+h_L}{h_{SL}{h}_L}}}.$$

(28)

Similarly, for the right shock, we obtain:

$$u_{SR}=u_R+\left(h_{SR}-h_R\right)D_4$$

(29)

and

$${\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}=D_4-g{\displaystyle \frac{\left(h_{SR}-h_R\right)}{4D_4{h}_{SR}^2}}$$

(30)

where

$$D_4=\sqrt{{\displaystyle \frac{1}{2}}g{\displaystyle \frac{h_{SR}+h_R}{h_{SR}{h}_R}}}$$

(31)

Rewriting Equations (6) and (7), we have

$$f_1\left(h_{LS},h_{RS}\right)=h_{SL}{u}_{SL}-h_{SR}{u}_{SR}=0$$

(32)

and

$$f_2\left(h_{LS},h_{RS}\right)=h_{SL}{u}_{SL}^2+{\displaystyle \frac{1}{2}}g{h}_{SL}^2-h_{SR}{u}_{SR}^2-{\displaystyle \frac{1}{2}}g{h}_{SR}^2-D=0$$

(33)

where u_SL is given by Equations (9) and (26), u_SR is given by Equations (12) and (29), and D is given by Equation (8). Taking the derivative of Equation (32) with respect to h_SL,

$${\displaystyle \frac{\partial f_1}{\partial h_{SL}}}=u_{SL}+h_{SL}{\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}.$$

(34)

Differentiating Equation (32) with respect to h_SR, we obtain:

$${\displaystyle \frac{\partial f_1}{\partial h_{SR}}}=-u_{SR}-h_{SR}{\displaystyle \frac{\partial u_{SR}}{h_{SR}}}$$

(35)

where u_SR is given by Equations (12) and (29), and ${\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}$ is given by Equations (14) and (30). Differentiating Equation (34) with respect to h_SL gives:

$${\displaystyle \frac{\partial f_2}{\partial h_{SL}}}=u_{SL}^2+2h_{SL}{u}_{SL}{\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}+g{h}_{SL}-g\Delta z$$

(36)

where u_SL is given by Equations (9) and (26), ${\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}$ is given by Equations (11) and (27), and $\Delta z=z_R-z_L$. Differentiating Equation (33) with respect to h_SR, we obtain

$${\displaystyle \frac{\partial f_2}{\partial h_{SR}}}=-u_{SR}^2-2h_{SR}{u}_{SR}{\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}-g{h}_{SR}$$

(37)

where u_SR is given by Equations (12) and (29), and ${\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}$ is given by Equations (14) and (30).

Table 2. New equations for Riemann solver of wet-bed SRP.

	Left		Right
	$u_{SL}$	${\displaystyle \frac{\partial u_{SL}}{\partial h_{SL}}}$	$u_{SR}$	${\displaystyle \frac{\partial u_{SR}}{\partial h_{SR}}}$
rarefaction	(9)	(11)	(12)	(14)
shock	(26)	(27)	(29)	(30)
	${\displaystyle \frac{\partial f_1}{\partial h_{SL}}}$$, {\displaystyle \frac{\partial f_2}{\partial h_{SL}}}$		${\displaystyle \frac{\partial f_1}{\partial h_{SR}}}$$, {\displaystyle \frac{\partial f_2}{\partial h_{SR}}}$
Main equation	$f_1\left(h_{SL},h_{SR}\right)$$, f_2\left(h_{SL},h_{SR}\right)$

summarizes the equations derived for the Riemann states in the SRP.

In the numerical model, we use the Levenberg–Marquardt method [41] (the source code can be downloaded from the website of the Utah University) to solve for h_SL and h_SR by configuring the main equation and Jacobi matrix according to Table 2. Then, u_SL and u_SR and the wave speed are determined either side of the step.

From the menu of functions available in the original code, we choose the f02 function which is called in subroutine lmder1_test and configure it with f₁ Equation (32) and f₂ Equation (33) as below:

fvec(1) = h1*uls − h2*urs;

fvec(2) = h1*uls*uls + 0.5*g*h1*h1 − h2*urs*urs − 0.5*gr*h2*h2 − r;

in which uls is u_SL, urs is u_SR, r is D, h1 is h_SL, and h2 is h_SR as given in Equations (32) and (33). The uls, urs and r variables are determined from Equations (8), (9), (11), (12), (14), (26), (27), (29), and (30) and the code inserted in the subroutine given in lmder1_test. The last 2 variables, h1 and h2, are unknowns that are then determined.

4. Results

Our proposed Riemann solver for the wet-bed SRP is tested for four benchmark cases involving different combinations of rarefactions and shock-like hydraulic bores in a straight, open channel containing a sudden step. The set-up and numerical results for each case are described below.

4.1. Case 1: Rarefaction-Static Shock-Rarefaction

This first case investigates flow over a 1 m high step within an otherwise horizontal channel of total length 25 m. The step is located at the middle of the channel and extends downstream, such that the bed elevation z = 0 for x < 12.5 m and z = 1 m for x ≥ 12.5 m.

Table 3. Initial conditions for rarefaction-static shock-rarefaction.

	h (m)	q (m2/s)	z (m)
Left (12.5 > x ≥ 0 m)	8.0	−16.0	0
Right (25 ≥ x > 12.5 m)	5.0	35.852	1

lists the initial conditions. Please note that the initial values of h_SL and h_SR at x = 12.5 m are both set to 5.0 m. The converged results obtained using LM iteration are h_SL = 4.39958 m, u_SL = 2.576215 m/s, h_SR = 2.92406 m, and u_SR = 3.87621 m/s. Figure 3 compares our numerical solution obtained on a grid of 250 cells at time t = 0.79932 s with the exact analytical solution derived by Murillo and García-Navarro [42]. The agreement is excellent, except for slight discrepancies at the smoothed ends of the rarefaction waves.

4.2. Case 2: Shock-Static Shock-Shock

The second case examines a tidal bore encountering a river flow.

Table 4. Initial conditions for shock-static shock-shock.

	h (m)	q (m2/s)	z (m)
Left (12.5 > x ≥ 0 m)	4.0	19	0
Right (25 ≥ x > 12.5 m)	1.0838	−2.3685	1

lists the initial conditions: on the left side of the step interface, there is a flow moving to the right with a depth of 4 m and a velocity of 4.75 m/s, while on the right side, the flow moves to the left with a depth of 1.0838 m and a speed of 2.1854 m/s.

The initial values of h_SL and h_SR are again set to 5.0 m. After application of LM iteration, the converged results are h_SL = 5.19336 m, u_SL = 2.99267 m/s, h_SR = 3.71648 m, and u_SR = 4.18192 m/s. Figure 4 compares the converged free-surface elevation profile along the channel on a grid of 250 grid cells at time t = 1.48842 s with the exact solution obtained by Murillo and García-Navarro [42]. Again, the two solutions are almost identical, except for a slight smoothing effect on the front face of the leftward propagating bore at x~7 m.

4.3. Case 3: Rarefaction-Static Shock-Shock

This scenario involves a dam-break event occurring over a wet bed, with the water in the open channel initially at rest. The channel is 25 m in length and is divided into 250 computational cells.

Table 5. Initial conditions for dam break in a channel with a step.

	h (m)	q (m2/s)	z (m)
Left (12.5 > x ≥ 0 m)	5.0	0	0
Right (25 ≥ x > 12.5 m)	1.0	0	1

provides the starting flow parameters on both sides of the step interface.

Our results converge to h_SL = 3.64399 m, u_SL = 2.08243 m/s, h_SR = 2.24788 m, and u_SR = 3.32036 m/s. Figure 5 shows the outstanding agreement obtained between the predicted profile by the present numerical scheme at t = 1.30815 s and the exact solution given by Murillo and García-Navarro [42].

4.4. Case 4: Comparison Between Present Algorithm and That of Xu et al. [38]

The foregoing classical cases do not distinguish between the present algorithm and that previously proposed by Xu et al. [38], with both algorithms giving the same results after several iterative steps, and the present method requiring 1 or 2 more steps. To examine the robustness of the new scheme, we consider the following case where h_L = 0.142500 m, u_L = 1.263158 m/s, h_R = 0.148000 m, u_R = 1.216216 m/s, and dz = 0.005500 m. We found that Xu et al.’s [38] algorithm was unable to achieve convergence whereas the present method provided the correct result: h_SL = 0.14912 m, u_SL = 1.20887 m/s, h_SR = 0.14925 m, and u_SR = 1.22638 m/s. This case demonstrates that the method proposed herein is more robust than the previous algorithm suggested by Xu et al. [38]. The exact solution given by Murillo and García-Navarro [42] is h_SL = 0.14525 m, u_SL = 1.263158 m/s, h_SR = 0.1425 m, u_SR = 1.263158 m/s.

5. Conclusions

This study has derived a pair of governing equations whose two dependent variables comprise local depths to the immediate left and right of an abrupt step in a one-dimensional wet-bed open channel. The equations are unified in representing combined shock and rarefaction conditions and are solved using the iterative Levenberg–Marquardt method. The new approach is simpler and more robust than previous methods and is demonstrated to give correct predictions of rarefaction and shock-like hydraulic bores for three classic benchmark and one realistic cases. The excellent agreement between the present model predictions and established analytical solutions demonstrates that our algorithm performs effectively over the range of cases considered herein. In future work, we intend to carry out a thorough comparison between our algorithm and previous numerical schemes in order to evaluate the relative errors and CPU run times on the same platform. It is also intended to extend the present algorithm to dry-bed cases and two-horizontal dimensions in space for the model to be applicable to more general simulations of practical engineering interest.