Abstract
This paper presents numerical modeling and investigation for the Ripa system. This model is derived from a shallow water model by merging the horizontal temperature gradients. We applied the non-homogeneous Riemann solver (NHRS) method for solving the Ripa model. This scheme contains two stages named predictor and corrector. The first one is made up of a control parameter that is responsible for the numerical diffusion. The second one recuperates the balance conservation equation. One of the main features of the NHRS scheme, it can determine the numerical flux corresponding to the real state of solution in the non-attendance of Riemann solution. Various test cases of physical interest are considered. These case studies display the high resolution of the NHRS scheme and emphasize its ability to produce accurate results for the Ripa model. The presented solutions are very critical in superfluid applications of energy and many others. Finally, the NHRS technique can be used to solve a wide range of additional models in applied research.