Positivity and bound preserving well-balanced high order compact finite difference scheme for Ripa and pollutant transport model

We construct a fourth-order accurate compact finite difference scheme that is well-balanced, positivity-preserving of water height, and bound-preserving of temperature for Ripa and concentration for pollutant transport systems. The proposed scheme preserves the still-water steady state and the positivity of water height. It also maintains concentration bounds for pollutants across nonflat bottom topographies, regardless of the presence of a pollutant source. Our approach incorporates water height and pollutant concentration constraints within the same discretization, utilizing weak monotonicity and a simple bound-preserving limiter while preserving the well-balanced property. Through extensive numerical simulations encompassing Ripa and pollutant transport models, we demonstrate the effectiveness of our method, verifying its well-balanced property, high-order accuracy, positivity-preserving, and bound-preserving capabilities.