Publications

Compact list of publications and talks

Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

Published in arXiv, 2022

In this paper, we study different high order FEM methods for hyperbolic problems, providing parameters that lead to stable and reliable schemes for triangular meshes. Download paper

Recommended citation: Michel, S., Torlo, D., Ricchiuto, M. and Abgrall, R.. Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping. arXiv:2206.06150 (2022). https://arxiv.org/abs/2206.06150

High order entropy preserving ADER scheme

Published in arXiv preprint, 2022

In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations. Download paper

Recommended citation: E. Gaburro, P. Öffner, M. Ricchiuto and D. Torlo. (2022). "High order entropy preserving ADER scheme. " arXiv preprint, 2022. https://arxiv.org/abs/2206.03889. https://arxiv.org/abs/2206.03889

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation

Published in arXiv preprint, 2022

We introduce arbitrary high order WENO finite volume schemes with global fluxes. The global flux includes the integral of the source term, so that it is natural to balance the moving equilibria for this kind of schemes. We show for shallow water equations with bathymetry that we can exactly preserve the discharge for moving steady states. Morover, we can apply a correction to be also well-balanced with respect to the lake at rest steady state. Download paper

Recommended citation: M. Ciallella, D. Torlo and M. Ricchiuto. (2022). "Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation. " arXiv preprint, 2022. https://arxiv.org/abs/2205.13315. https://arxiv.org/abs/2205.13315

Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme

Published in Communications in Mathematical Sciences, 2022

In this short paper, we intend to describe one way to construct arbitrarily high order kinetic schemes on regular meshes. The method can be arbitrarily high order in space and time, run at least CFL one, is asymptotic preserving and computationally explicit, i.e., the computational costs are of the same order of a fully explicit scheme. We also introduce a nonlinear stability method that enables to simulate problems with discontinuities, and it does not kill the accuracy for smooth regular solutions. Download paper

Recommended citation: R. Abgrall and D. Torlo. (2022). "Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme. " Communications in Mathematical Sciences, 20, 2, 297-326. https://dx.doi.org/10.4310/CMS.2022.v20.n2.a1. https://dx.doi.org/10.4310/CMS.2022.v20.n2.a1

Model order reduction strategies for weakly dispersive waves

Published in arXiv, 2021

Water waves can be approximated with different models. Dispersive-hyperbolic models serve this scope under smallness conditions of nonlinearity and shallowness parameters. The discretization of these models consists often of a hyperbolic system coupled with an elliptic system. In this work we reduce with standard model order reduction techniques the elliptic operator. Finally, we apply some hyperreduction to reduce the whole system. Download paper

Recommended citation: D. Torlo and M. Ricchiuto. (2021). "Model order reduction strategies for weakly dispersive waves. " arXiv preprint, https://arxiv.org/abs/2112.10608. https://arxiv.org/abs/2112.10608

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Published in arXiv, 2021

In shallow water equations simulations the positivity of water height is a fundamental property to preserve. We use a linearly implicit modified Patankar Deferred Correction method to guarantee its positivity without any restriction on the time step. The rest of the discretization is obtained with a classical WENO5 finite volume method. Download paper

Recommended citation: M. Ciallella, L. Micalizzi, P. Öffner and D. Torlo. (2021). "An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations. " arXiv preprint, https://arxiv.org/abs/2110.13509. https://arxiv.org/abs/2110.13509

Analytical traveling vortex solutions of hyperbolic equations for validating very high order schemes

Published in arXiv, 2021

Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like analytical solutions in 2 dimensions for the shallow water equations (and, hence, Euler equations) that can be used to test the order of accuracy of numerical methods. These solutions have different smoothness in their derivatives (up to arbitrary derivatives) and can be used accordingly to the order of accuracy of the scheme to test. Download paper

Recommended citation: M. Ricchiuto and D. Torlo. (2021). "Analytical traveling vortex solutions of hyperbolic equations for validating very high order schemes. " arXiv preprint, https://arxiv.org/abs/2109.10183. https://arxiv.org/abs/2109.10183

Issues with Positivity-Preserving Patankar-type Schemes

Published in arXiv, 2021

We study various properties for a class of positivity-preserving nonlinear schemes (Patankar-type schemes) and we discover two types of issues: oscillations around stady states when the timestep is too large and spurious steady states where some methods get stuck. Download paper

Recommended citation: D. Torlo, P. Öffner and H. Ranocha. (2021). "Issues with Positivity-Preserving Patankar-type Schemes. " arXiv preprint, https://arxiv.org/abs/2108.07347. https://arxiv.org/abs/2108.07347

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

Published in arXiv, 2021

In this paper, we study different high order FEM methods for hyperbolic problems, providing parameters that lead to stable and reliable schemes. Download paper

Recommended citation: R. Abgrall, E. Le Mélédo, P. Öffner and D. Torlo. (2021). "Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes. " arXiv preprint, https://arxiv.org/abs/2106.05005. https://arxiv.org/abs/2106.05005

Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

Published in arXiv, 2021

In this paper, we study different high order FEM methods for hyperbolic problems, providing parameters that lead to stable and reliable schemes. Download paper

Recommended citation: Michel, S., Torlo, D., Ricchiuto, M. and Abgrall, R.. Spectral Analysis of Continuous FEM for Hyperbolic PDEs: Influence of Approximation, Stabilization, and Time-Stepping. J Sci Comput 89, 31 (2021). https://doi.org/10.1007/s10915-021-01632-7 https://doi.org/10.1007/s10915-021-01632-7

DeC and ADER: Similarities, Differences and a Unified Framework

Published in Journal of Scientific Computing, 2021

In this paper, we demonstrate that the explicit ADER approach can be seen as a special interpretation of the deferred correction (DeC) method. Download paper

Recommended citation: M. H. Veiga, P. Öffner, and D. Torlo. (2021). "DeC and ADER: Similarities, Differences and a Unified Framework." Journal of Scientific Computing, 87, 2 (2021). https://doi.org/10.1007/s10915-020-01397-5. https://doi.org/10.1007/s10915-020-01397-5

Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

Published in Applied Numerical Mathematics, 2020

Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Download paper

Recommended citation: P. Öffner and D. Torlo. (2020). "Arbitrary high-order, conservative and positivity preserving Patankar--type deferred correction schemes." Applied Numerical Mathematics, 153:15 – 34. https://doi.org/10.1016/j.apnum.2020.01.025

High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

Published in SIAM Journal on Scientific Computing, 2020

This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. Download paper

Recommended citation: R. Abgrall, and D. Torlo. (2020). "High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models." SIAM Journal on Scientific Computing, 42(3): B816--B845. https://doi.org/10.1137/19M128973X

Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases

Published in arXiv, 2020

In this work, we study MOR algorithms for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. Download paper

Recommended citation: D. Torlo. (2020). "Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases." arXiv preprint, arXiv:2003.13735. https://arxiv.org/abs/2003.13735

Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification

Published in Journal of Computational and Applied Mathematics, 2019

In this work, we present a model order reduction (MOR) technique for hyperbolic conservation laws with applications in uncertainty quantification (UQ). Download paper here

Recommended citation: R. Crisovan, D. Torlo, R. Abgrall, and S. Tokareva. (2019). "Model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification." Journal of Computational and Applied Mathematics, 348:466 – 489. https://doi.org/10.1016/j.cam.2018.09.018

Published in , 1900

Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs

Published in SIAM/ASA Journal on Uncertainty Quantification, 2018

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. Download paper

Recommended citation: D. Torlo, F. Ballarin, and G. Rozza. (2018). "Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs." SIAM/ASA Journal on Uncertainty Quantification, 6(4): 1475--1502. https://doi.org/10.1137/17M1163517