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
Recent Works
Wasilij Barsukow, Mirco Ciallella, Mario Ricchiuto, and Davide Torlo. Genuinely multi-dimensional stationarity preserving finite volume formulation for nonlinear hyperbolic pdes. Journal of Computational Physics, 550:114633, 2026.
Bender, J., Izgin, T., Offner, Ph. and Torlo, D. (2026), The Lax–Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws. Computers and Fluids, 304: 106885, 2026.
Barsukow, W., Ricchiuto, M. and Torlo, D. (2025), Structure Preserving Nodal Continuous Finite Elements via Global Flux Quadrature. Numer Methods Partial Differential Eq., 41: e23167. https://doi.org/10.1002/num.23167
M. Ciallella, L. Micalizzi, V. Michel-Dansac, P. Offner and D. Torlo. "A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations." Int J Geomath 16, 6 (2025). https://doi.org/10.1007/s13137-025-00262-7
E. Gaburro, P. Öffner, M. Ricchiuto and D. Torlo. "High order entropy preserving ADER-DG scheme." Applied Mathematics and Computation, 440:127644, 2023. https://doi:10.1016/j.amc.2022.127644
M. Ciallella, D. Torlo and M. Ricchiuto. "Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation. " Journal of Scientific Computing 96, 53 (2023). https://doi.org/10.1007/s10915-023-02280-9
M. Ciallella, L. Micalizzi, P. Öffner and D. Torlo. (2022). "An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations. " Computers & Fluids, 247, page 105630.
D. Torlo, P. Öffner and H. Ranocha. (2022). "Issues with Positivity-Preserving Patankar-type Schemes. " Applied Numerical Mathematics, 182, 117-147.