Introduction: The shallow water equations are a set of hyperbolic balance laws that describe the behavior of water flow in shallow regions such as rivers, lakes, and oceans. Solving hyperbolic balance laws poses significant challenges due to the presence of non-conservative terms, shocks and discontinuities. Analytical solutions are limited to simplified cases, so numerical methods are often employed to solve these equations. Numerical schemes addressing these balance laws must ensure the well-balanced property (Bermudez and Vázquez 1994), ensuring that discretized numerical fluxes must exactly balance by the approximated source terms. These types of numerical schemes utilize upwind/flux splitting techniques to handle wave propagation and discontinuities. Such well-balanced approaches work well for supercritical or subcritical regions but are known to struggle when Riemann problem includes both (LeFloch and Thanh 2011)- particularly in trans-critical flows and hydraulic jumps. To address this, various treatments, such as entropy fixes, shock fitting techniques, have been developed. Notably, Akbari and Pirzadeh (2022) (Akbari and Pirzadeh, 2022)introduced a set of shockwave fixes to cure the numerical slowly moving shock anomaly. Their approach is advantageous in accurately capturing the hydraulic jump. However, such scheme is only first-order accurate, as higher-order schemes progress, it becomes necessary to extend such technique to greater accuracy in high-resolution schemes.
Methodology: A second order well balanced numerical scheme has been designed for the shallow water equations using a semi-discrete MUSCL reconstruction. The first step in the semi-discrete finite volume method is to discretize the governing equations in space. For the one-dimensional shallow water equations, this involves dividing the computational domain into a set of control volumes and approximating the integral form of the conservation equations over each control volume. By considering the fluxes at the control volume interfaces and accounting for the source terms, a system of ordinary differential equations (ODEs) can be obtained. To ensure accurate and stable solutions, a second-order finite volume approach is employed for spatial discretization. The proposed approach aims to exactly preserve all steady states of shallow water equations while maintaining the second order of accuracy. To achieve this, we extend a recently developed fully well-balanced scheme, called HLL-MSF, to higher-order of accuracy. To upgrade the first-order HLL-MSF
 
scheme to second order while maintaining the same well-balanced property of the first order one, a MUSCL reconstruction approach with a suitable weighted technique is proposed. The weighted approach allows the numerical scheme to revert to the first order scheme with shockwave fixes at hydraulic jumps or at trans-critical points. Appropriate flux limiters are also introduced to ensure the well-balanced property of the numerical scheme in smooth steady state cases. The method's accuracy and stability are attributed to these carefully chosen flux limiters and weighted coefficients. The final step in the semi-discrete finite volume method involves time integration to advance the solution in time. In this paper, the third order explicit Runge-Kutta method is chosen as the time integration scheme. By combining the second-order finite volume spatial discretization and the third-order explicit Runge-Kutta time integration scheme, the proposed finite volume method ensures higher-order accuracy in both space and time.
Results and Discussion: To verify the well-balanced property and the second order of accuracy of the proposed numerical scheme several numerical examples and benchmarks found in the literature including both steady and unsteady cases are presented. For numerical experiments that have analytical or reference solutions, numerical errors are calculated using L¹ and L^∞ norms. The first test case is devoted to the simulation of steady state at rest or the lake at rest situation. Numerical errors demonstrate that the proposed scheme is exactly well-balanced in this case. The second test case addresses a smooth steady state of trans-critical flow over a bump. The proposed second order scheme is confirmed to capture the smooth steady state precisely (Table 1). We also perform experiments on trans-critical flow with hydraulic jump to see how the proposed scheme behaves when the solution contains a shock discontinuity. Unlike the traditional higher-order schemes which often use the pre-balanced shallow water formulation to achieve the exact conservation property on steady state cases at rest, the proposed second order scheme can capture both smooth and non-smooth (Hydraulic jump) parts exactly with no smears and oscillations (Table 1). An additional test case is conducted to confirm the second order accuracy of the numerical scheme. Table 2 Illustrates that the intended accuracy is clearly achieved. Finally, three numerical experiments are conducted in quasi-steady and unsteady conditions including slowly moving shocks over flat or discontinuous topography. The higher-order approximate solvers are known to achieve better accuracy for such flows than the first order counterparts.
Conclusion: In this paper, second-order well-balanced numerical schemes are developed for the solution of one-dimensional shallow water equations. The approach accurately models different regimes of the flow accurately. The advantage of the proposed scheme over existing higher-order schemes is the fully well-balanced and entropy satisfying properties, where all steady states solutions are exactly preserved.