Davide Torlo

Rémi Abgrall, Davide Torlo

This work aims to extend the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve hyperbolic system of partial differential equations. Up to our knowledge, it was used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an IMEX (implicit--explicit) version of the residual distribution schemes, that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g. multiphase flows, kinetic models, viscoelasticity problems. Moreover, the provided scheme is able to catch different relaxation scales automatically, without losing accuracy. The scheme is asymptotic preserving and this guarantees that in the relaxation limit, we recast the expected macroscopic behaviour. To get a high order accuracy, we use an IMEX time discretization combined with a Deferred Correction (DeC) procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in 1D and 2D for stiff systems of equations.

Moussa Ziggaf, Davide Torlo, Mario Ricchiuto

We develop arbitrarily high-order, stationarity-preserving stabilized finite element methods for multidimensional nonlinear hyperbolic balance laws on Cartesian grids. We aim at approximating all the steady states of the problem at hand, including non-trivial genuinely multidimensional equilibria, with a level of accuracy higher than the nominal one of the underlying scheme. We formalize more precisely the meaning of stationarity preservation, providing some technical conditions for its realizability. We then recast the multidimensional global-flux quadrature of Barsukow et al. (Num. Meth. PDEs, 2025) as a local preprocessing of the physical fluxes that maps continuous polynomial vector fields to a local space with Raviart--Thomas-type structure. Both the Galerkin and SUPG formulations are recast in this setting. The resulting methods extend the stationarity-preserving finite-volume approach of Barsukow et al. (J. Comput. Phys., 2026) to high-order continuous finite elements and Barsukow et al. (Num. Meth. PDEs, 2025) to nonlinear balance laws. We analyze key properties of the proposed schemes, including local conservation and nodal superconvergence of the discrete steady kernel, and we discuss their relation to low-Mach-compliant discretizations. We apply the framework to the compressible Euler equations with gravity. A simple source-term reformulation yields machine-precision preservation of isothermal hydrostatic equilibria. Extensive numerical benchmarks, including moving equilibrium, near-equilibrium, and instability-dominated regimes, demonstrate clear improvements in robustness and accuracy over standard SUPG and reference finite-volume methods.

Davide Torlo, Francesco Ballarin, Gianluigi Rozza

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline--Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.

Luca Venturi, Davide Torlo, Francesco Ballarin et al.

In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.