Numerical methods that approximate the solutions of differential equations are not always able to preserve secondary physical properties indirectly derived from the mathematical model. The positivity of certain quantities is an example of such properties, which can be observed in many equations, e.g., density and pressure in compressible flows, water height in shallow water equations, concentrations of chemical reactants, population densities, and so on. Patankar-type schemes are particularly effective in preserving the positivity of these quantities.
Another important property in many PDEs is the preservation of equilibrium states. Ensuring that numerical methods preserve these equilibria (either exactly or with higher accuracy than the method’s standard order) leads to more reliable and cost-effective schemes. Such methods achieve significantly more precise simulations near equilibrium states and are capable of maintaining steady states over long periods. 

Components: Gabriella Puppo and Davide Torlo