A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography

Abstract

Available online xxxxKeywords: Two-layer system Well-balanced model Nonconservative 2LSWE HLL A robust and well-balanced numerical model is developed for solving the two-layer shallow water equations based on the approximate Riemann solver in the framework of finitevolume methods. The HLL (Harten, Lax, and van Leer) solver is employed to calculate the numerical fluxes. The numerical balance between the flux gradient and the source terms is achieved by using a balance-reformulation method. To obtain exactly the lake-atrest solutions as the water depth is chosen as the conserved variable for the continuity equations, a modified HLL flux formulation is proposed for mass flux calculations. Several numerical tests used to validate the performance of the developed numerical model. The results show that the developed model is accurate, well balanced, and that it predicts no oscillations around large gradients.

A Two‐Layer Hydrostatic‐Reconstruction Method for High‐Resolution Solving of the Two‐Layer Shallow‐Water Equations over Uneven Bed Topography

“…To purify the problem, as was done in previous relevant studies (e.g., Lee et al [9]; Swartenbroekx et al [20]; Hu et al [21]), the flows in the layers are assumed to be immiscible and have constant densities in each layer. Following Lu et al [11], we consider solving the following 2LSWEs system:…”

“…This method is able to numerically maintain quiescent flow at rest, yet it violates the principle of mass conservation because the temporal variation of the interface is unconsidered. An alternative ad-hoc technique 2 Mathematical Problems in Engineering was proposed by Lu et al [11], by limiting the diffusion part of numerical fluxes so that a zero mass flux is predicted under a QFC. To achieve high-resolution numerical solutions, Abgrall and Karni [12] added two auxiliary equations to the conventional 2LSWEs with a relaxation parameter; their numerical results showed that this method generally works well, but spurious oscillation may occur when an inappropriate relaxation parameter is chosen and the predictions may be grid-dependent.…”

Convergence Improved Lax-Friedrichs Scheme Based Numerical Schemes and Their Applications in Solving the One-Layer and Two-Layer Shallow-Water Equations

“…wherẽ− 1/2 is calculated by (17); the minimum is defined over the whole simulation domain. Note that, for this particular test, is commonly set to be a large value (e.g., ≈ 0.5) in literatures [8] and thus the numerical diffusion in the predictions is insignificant.…”

“…− ℎ 1 ( ℎ 2 / ) in (24) is a nonconservative product term, which has no definition around shocks and discontinuities and is tough to process in numerical discretizations. Following Spinewine et al [18], and Lu et al [17], the 2LSWEs are reformulated to make the nonconservative product term vanish. This is done by replacing the equations governing the lower-layer flow motions by a combination of the equations for the two-layer flow system as a whole; namely, the 2LSWEs are recast to (10) with vectors defined by…”

“…As the 1LSWEs and 2LSWEs are written in the same conservative form (see (10)), the 2LSWEs are solved similarly as that for the 1LSWEs presented in Section 2.1, with a difference in calculating̃in (17). In solving the 2LSWEs, for both the FORCE and SLIC schemes,̃≡ Δ /Δ , and, for both the C-FORCE and C-SLIC schemes,̃is set to be the approximately maximum local characteristic speed as [17,18] the robustness and accuracy of numerical schemes in solving the 2LSWEs [19][20][21]. The configuration of this test is the same as that in the one-layer dam-break test presented in Section 2.2, but with an upper-layer superimposed on the lower-layer.…”