A smoothness indicator for numerical solutions to the Ripa model

The Ripa model is the system of shallow water equations taking the water temperature fluctuations into account. For one-dimensional case, the Ripa model consists of three partial differential equations relating to three primitive variables, namely water depth, velocity, and temperature. The Ripa model is hyperbolic, and its solution can be discontinuous. When the Ripa model is solved using a conservative numerical method, the solution is usually diffusive around discontinuities. The diffusion at rough regions (around discontinuities, such as contact and shock discontinuities) makes the solution inaccurate. In practice, we want to know the places where the solution is accurate, and where it is inaccurate. That is, we want to know where the solution is smooth, and where it is rough. In this paper we propose the numerical entropy production to detect the smoothness of numerical solutions to the Ripa model. Numerical results show that the numerical entropy production is a robust smoothness indicator for numerical solutions to the Ripa model.