Well-balanced and positivity-preserving wet-dry front reconstruction scheme for Ripa models

This paper explores the reconstruction of wet-dry fronts (WDF) for solving both one-dimensional (1D) and two-dimensional (2D) Ripa systems, with a particular emphasis on the influence of temperature. Our aim is to develop a well-balanced numerical scheme that not only preserves the steady state but also ensures the positivity of both water height and temperature. By employing conservative variables for reconstruction instead of equilibrium variables, we have achieved a significant doubling of the CFL number for fully flooded cells. We have refined the original 1D WDF reconstruction method and further enhanced the corresponding 2D scheme. The conservation principle and linearity observed in the wet region of partially flooded cells indicate a constant cell-wise velocity and temperature. Additionally, we introduce a novel draining time approach to adjust the numerical flux in an upwind manner, ensuring both stability and efficiency, even for partially flooded cells. Numerical examples are presented to demonstrate the well-balanced property, high-order accuracy, and positivity-preserving characteristics of our proposed method. These examples also showcase the method's ability to capture small perturbations in the lake-at-rest steady state, highlighting its potential for practical applications.