High Order Schemes for Conservation laws 

Many phenomena in science are described by means of systems of nonlinear hyperbolic balance laws. They include gas dynamics, shallow water flows, plasmas. These equations are usually not solvable analytically. It is thus necessary to design robust numerical schemes to compute their approximate solutions. We work on the development and study of efficient and high-order accurate finite volume, finite element and discontinuous Galerkin methods for the numerical evolution of hyperbolic PDEs, such as CWENO schemes and their implicit formulation (Quinpi), ADER-DG schemes and Deferred Correction schemes, and high order schemes based on Lagrange Galerkin methods for Fokker Plank equations.

Components: Elisabetta Carlini, Gabriella Puppo, Davide Torlo and Giuseppe Visconti

Recent works

• M. Briani, G. Puppo, G. Visconti. Dissipation-dispersion analysis of fully-discrete implicit discontinuous Galerkin methods and application to stiff hyperbolic problems. Numer. Methods Partial Differential Equations, 42(1). 2026

• W. Barsukow, M. Ricchiuto, and D. Torlo. Structure Preserving Nodal Continuous Finite Elements via Global Flux Quadrature. Numer Methods Partial Differential Eq., 41: e23167. 2025. https://doi.org/10.1002/num.23167 

• S. Gerster, A. Sikstel, G. Visconti. Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function. Numer. Math., 2025.

• Öffner, P., Petri, L., and Torlo, D.  Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations. Applied Numerical Mathematics, 212, p. 110-134, 2025.

• 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.

• G. Puppo, M. Semplice, G. Visconti. Quinpi: Integrating stiff hyperbolic systems with implicit high order finite volume schemes. Commun. Comput. Phys., 36(1):30-70. 2024.

• M. Han Veiga, L. Micalizzi and D. Torlo. On improving the efficiency of ADER methods. Applied Mathematics and Computation, 466, page 128426, 2024.

• G. Puppo, M. Semplice, G. Visconti. Quinpi: integrating conservation laws with CWENO implicit methods. Commun. Appl. Math. Comput., 5:343-369. 2023.

• L. Micalizzi, D. Torlo and W. Boscheri. Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order. Commun. Appl. Math. Comput. 2023.

• L. Micalizzi and D. Torlo. "A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity. " Commun. Appl. Math. Comput. 2023. https://doi.org/10.1007/s42967-023-00294-6