# Sitemap

A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.

## Davide Torlo's Personal Website

I am an Assistant Professor in Numerical Analysis, focusing on hyperbolic PDEs and model order reduction.

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## How long will it take to adopt a Zero Covid Strategy?

Published:

In this post, I try to get further information on how long does for a Zero Covid strategy to get to the end of the first phase and I provide a tool to play with the parameters on Google colab.

## How is testing going for different World countries for the SARS-CoV-2 pandemic

Published:

This post is based on the observation I made some months ago in a Twitter post.

## Weighted Model Order Reduction to Quantify Uncertainties

Published:

This is a blog post I wrote for the SISSA mathLab medium page.

## Portfolio item number 1

Short description of portfolio item number 1

## Portfolio item number 2

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## 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.

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
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## Weighted reduced order methods for parametrized partial differential equations with random inputs

Published in Uncertainty Modeling for Engineering Applications, PoliTO Springer Series, 2019

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.

Recommended citation: Venturi, L., Torlo, D., Ballarin, F., Rozza, G. (2019). " Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs. " In: Canavero, F. (eds) Uncertainty Modeling for Engineering Applications. PoliTO Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-030-04870-9_2 https://doi.org/10.1007/978-3-030-04870-9_2
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## 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).

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
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## 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.

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
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## 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.

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
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## 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.

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
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## 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.

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

Published in Journal of Scientific Computing, 2021

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

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
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## Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

Published in The SMAI Journal of computational mathematics, 2021

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

Recommended citation: R. Abgrall, E. Le Mélédo, P. Öffner and D. Torlo. (2022). "Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes. " The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 125-160. doi:10.5802/smai-jcm.82 https://doi.org/10.5802/smai-jcm.82
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## Issues with Positivity-Preserving Patankar-type Schemes

Published in Applied Numerical Mathematics, 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.

Recommended citation: D. Torlo, P. Öffner and H. Ranocha. (2022). "Issues with Positivity-Preserving Patankar-type Schemes. " Applied Numerical Mathematics, 182, 117-147. https://doi.org/10.1016/j.apnum.2022.07.014
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## 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.

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

Published in Computer and Fluids, 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.

Recommended citation: 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. https://doi.org/10.1016/j.compfluid.2022.105630
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## Model order reduction strategies for weakly dispersive waves

Published in Mathematics and Computers in Simulation, 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.

Recommended citation: D. Torlo and M. Ricchiuto. "Model order reduction strategies for weakly dispersive waves. " Mathematics and Computers in Simulation, (205), pages 997-1028, 2023. https://doi.org/10.1016/j.matcom.2022.10.034
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## 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.

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

Published in Journal of Scientific Computing, 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.

Recommended citation: M. Ciallella, D. Torlo and M. Ricchiuto. (2022). "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. https://doi.org/10.1007/s10915-023-02280-9
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## High order entropy preserving ADER-DG scheme

Published in Applied Mathematics and Computation, 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.

Recommended citation: E. Gaburro, P. Öffner, M. Ricchiuto and D. Torlo. "High order entropy preserving ADER-DG scheme." Applied Mathematics and Computation, 440:127644, 2023. doi:10.1016/j.amc.2022.127644. https://doi.org/10.1016/j.amc.2022.127644
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## Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping

Published in Journal of Scientific Computing, 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.

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. J Sci Comput 94, 49 (2023). https://doi.org/10.1007/s10915-022-02087-0
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## Weighted Reduced Order Methods for Uncertainty Quantification

Published in Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, chapter 12, pages 251-264. Society for Industrial & Applied Mathematics, U.S., 2022

Partial differential equations (PDEs) are an effective tool to model phenomena in applied sciences. Realistic problems usually depend on several physical and geometrical parameters that can be calibrated by using real data. In real scenarios, however, these parameters are affected by uncertainty due to measurement errors or scattered data information. To deal with more reliable models which take this issue into account, stochastic PDEs can be numerically approximated. In the uncertainty quantification (UQ) context, many simulations are run to better understand the system at hand and to compute statistics of outcomes over quantities of interest. In particular, the input parameters of the stochastic PDEs are assumed to be random finite-dimensional variables.

Recommended citation: Davide Torlo, Maria Strazzullo, Francesco Ballarin, and Gianluigi Rozza. Weighted reduced order methods for uncertainty quantification. In Francesco Ballarin Gianluigi Rozza, Giovanni Stabile, editor, Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics, chapter 12, pages 251-264. Society for Industrial & Applied Mathematics, U.S., 2022. https://doi.org/10.1137/1.9781611977257.ch12 https://doi.org/10.1137/1.9781611977257.ch12
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## A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity

Published in Commun. Appl. Math. Comput., 2022

Deferred Correction methods are arbitrarily high order methods that consists of an iterative procedure. At each iterations the high order reconstruction is updated leading to costs that scale as the square of the order of accuracy. We propose a way to cut up to half of the computational costs for this methods by increasing the order of the reconstruction at each iteration. An adaptive version allows also to set a priori a tolerance to reach a certain error. Applications to PDEs within the RD-DeC frameworks allows as well a great computational advantage.

Recommended citation: 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. https://doi.org/10.1007/s42967-023-00294-6
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## A necessary condition for non oscillatory and positivity preserving time-integration schemes

Published in arXiv preprint, 2022

Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate the properties of MP schemes. In Izgin et al. ESAIM: M2AN, 56 (2022), inspired from dynamical systems, the Lyapunov stability properties of such schemes have been investigated, while in Torlo et al. Appl. Numer. Math., 182 (2022) their oscillatory behaviour has been studied. In this work, we investigate the connection between the oscillatory behaviour and the Lyapunov stability and we prove that a condition on the Lyapunov stability function is necessary to avoid oscillations. We verify our theoretical result on several numerical tests.

Recommended citation: T. Izgin, P. Öffner and D. Torlo. "A necessary condition for non oscillatory and positivity preserving time-integration schemes." (2022) arXiv preprint, arXiv:2211.08905. https://arxiv.org/abs/2211.08905
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## An optimisation-based domain-decomposition reduced order model for the incompressible Navier-Stokes equations

Published in Computers and Mathematics with Applications, 2022

The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain-decomposition (DD) methods and reduced-order modelling (ROM). In particular, we consider an optimisation-based domain-decomposition algorithm for the parameter-dependent stationary incompressible Navier-Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal-control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward-facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain-decomposition algorithm.

Recommended citation: I. Prusak, M. Nonino, D. Torlo, F. Ballarin and G. Rozza. "An optimisation-based domain-decomposition reduced order model for the incompressible Navier-Stokes equations." Computers & Mathematics with Applications, 151 (2023) 172-189. https://doi.org/10.1016/j.camwa.2023.09.039
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## Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order

Published in Commun. Appl. Math. Comput., 2022

We optimize iterative arbitrary high order methods. The main idea, similar to the work on efficient DeC, is to match space discretization and accuracy of the iteration. This allows to have p-adaptivity very easily without wasting any computational step. We apply it to ADER-DG and ADER-FV with an a posteriori limiter: Discrete Optimally increasing Order Method (DOOM). It can provably preserve structures of the scheme such as positivity. Download paper

Recommended citation: 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). https://doi.org/10.1007/s42967-023-00290-w
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## On improving the efficiency of ADER methods

Published in Applied Mathematics and Computation, 2023

The arbitrary derivative (ADER) approach for numerical differential problem solution is enhanced in this study. Improvements include higher-order accuracy through precise polynomial discretization choices and integration with Deferred Correction (DeC) formalism. Analytical and numerical results cover stability analysis, computational efficiency, adaptivity, and hyperbolic PDE applications with Spectral Difference (SD) discretization. Download paper

Recommended citation: M. Han Veiga, L. Micalizzi and D. Torlo. "On improving the efficiency of ADER methods." Applied Mathematics and Computation, 466, page 128426, 2024. https://doi.org/10.1016/j.amc.2023.128426
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## An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems

Published in Computers & Mathematics with Applications, 2023

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the domain decomposition approach, we derive an optimal control problem, for which we present the convergence analysis. The snapshots for the high-fidelity model are obtained with the Finite Element discretisation, and the model order reduction is then proposed both in terms of time and physical parameters, with a standard POD-Galerkin projection. We test the proposed methodology on two fluid dynamics benchmarks: the non-stationary backward-facing step and lid-driven cavity flow. Finally, also in view of future works, we compare the intrusive POD–Galerkin approach with a non–intrusive approach based on Neural Networks.

Recommended citation: I. Prusak, D. Torlo, M. Nonino and G. Rozza. "An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems." (2024) Computers & Mathematics with Applications, 166, 253-268. https://doi.org/10.1016/j.camwa.2024.05.004
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## Friedrichs’ systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions

Published in arXiv preprint, 2023

Friedrichs’ systems unify various elliptic, parabolic and hyperbolic semi-linear PDEs. We use a discontinuous Galerkin discretization to approximate such equations and we apply classical model order reduction techniques for parameterized problems. To tackle the slow Kolmogorov n-width that they show, we adopt two techniques. First, we use a domain decomposition based on different indicators, that allow to reduce the number of basis on smooth areas. Secondly, inspired by the concept of vanishing viscosity, we adopt a Graph Neural Network that forecasts the solution given some high viscosity solutions for the same parameter obtained with the presented reduction techniques, recalling that for high viscosity we do not observe slow Kolmogorov n-width.

Recommended citation: F. Romor, D. Torlo and G. Rozza. "Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions." (2023) arXiv preprint, arXiv:2308.03378. https://arxiv.org/abs/2308.03378
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## Computations for Sustainability

Published in Quantitative Sustainability. Springer, Cham., 2024

Parallel to the need for new technologies and renewable energy resources to address sustainability, the emerging field of Artificial Intelligence (AI) has experienced continuous high-speed growth in the application of its capabilities of modelling, managing, processing, and making sense of data in the entire areas related to the production and management of energy. Moreover, the current trend indicates that the energy supply and management process will eventually be controlled by autonomous smart systems that optimize energy distribution operations based on integrative data-driven Machine Learning (ML) techniques or other types of computational methods.

Recommended citation: Salavatidezfouli, S., Nikishova, A., Torlo, D., Teruzzi, M., Rozza, G. (2024). Computations for Sustainability. In: Fantoni, S., Casagli, N., Solidoro, C., Cobal, M. (eds) Quantitative Sustainability. Springer, Cham. https://doi.org/10.1007/978-3-031-39311-2_7 https://doi.org/10.1007/978-3-031-39311-2_7
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## Optimisation–Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics

Published in arXiv preprint, 2024

In the context of domain decomposition optimization based models, we compare different couplings of full and reduced order models of each subcomponent.

Recommended citation: I. Prusak, D. Torlo, M. Nonino and G. Rozza. "Optimisation–Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics." (2024) arXiv preprint, arXiv:2402.10570. https://arxiv.org/abs/2402.10570
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## A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations

Published in arXiv preprint, 2024

We combine the modified Patankar Deferred Correction approach for the positivity of Shallow Water equations with a hydrostatic reconstruction technique to preserve global equilibria. We focus on tough tests aiming at flooding of urban areas.

Recommended citation: 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." (2024) arXiv preprint, arXiv:2402.12248. https://arxiv.org/abs/2402.12248
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## Calibration-Based ALE Model Order Reduction for Hyperbolic Problems with Self-Similar Travelling Discontinuities

Published in ArXiv, 2024

We propose a novel Model Order Reduction framework that is able to handle solutions of hyperbolic problems characterized by multiple travelling discontinuities. By means of an optimization based approach, we introduce suitable calibration maps that allow us to transform the original solution manifold into a lower dimensional one. The optimization process does not require the knowledge of the discontinuities location. In the online phase, the coefficients of the projection of the reduced order solution onto the reduced space are recovered by means of an Artificial Neural Network. To validate the methodology, we present numerical results for the 1D Sod shock tube problem and for the 2D double Mach reflection problem, also in the parametric case.

Recommended citation: Nonino, M., & Torlo, D. (2024). Calibration-Based ALE Model Order Reduction for Hyperbolic Problems with Self-Similar Travelling Discontinuities. arXiv preprint arXiv:2403.11664 . https://arxiv.org/abs/2403.11664
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## Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations

Published in ArXiv, 2024

We study the implicit and IMEX version of DeC and ADER time integration method with application to advection-diffusion and advection-dispersion equations. The study is based on stability analysis.

Recommended citation: Öffner, P., Petri, L., and Torlo, D. (2024). Analysis for Implicit and Implicit-Explicit ADER and DeC Methods for Ordinary Differential Equations, Advection-Diffusion and Advection-Dispersion Equations. arXiv preprint arXiv:2404.18626. https://arxiv.org/abs/2404.18626
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## A time-adaptive algorithm for pressure dominated flows: a heuristic estimator

Published in ArXiv, 2024

We compare some time-adaptive algorithms based on error estimators that use variations of BDF2 and BDF3 time marching schemes, for Navier-Stokes and FSI problems with pressure dominated flows.

Recommended citation: Prusak, I., Torlo, D., Nonino, M. and Rozza, G. (2024). A time-adaptive algorithm for pressure dominated flows: a heuristic estimator. arXiv preprint arXiv:2407.00428. https://arxiv.org/abs/2407.00428
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## Structure preserving nodal continuous Finite Elements via Global Flux quadrature

Published in ArXiv, 2024

We propose a Finite Element stabilized solver that is able to preserve multi-dimensional equilibria (div-free) using the Global Flux quadrature.

Recommended citation: Barsukow, W., Ricchiuto, M., and Torlo, D. (2024). Structure preserving nodal continuous Finite Elements via Global Flux quadrature. arXiv preprint arXiv:2407.10579. https://arxiv.org/abs/2407.10579
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## ADER and DeC: Arbitrarily High Order Explicit Time Integration Methods

Published:

When we think about high order time integration, Runge–Kutta (RK) is the most known class of schemes that comes to mind, for historical reasons. Nevertheless, many other techniques have been developed during these years to improve these techniques and to get a generalized form of them.

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

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Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems.

## Hyperbolic problems: high order methods and model order reduction

Published:

PhD Defense at UZH Zurich. The talk summarize the main topics of the thesis: high order methods for ODEs, hyperbolic PDEs and model order reduction techniques.

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

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Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. 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. Finally, we validate our theoretical analysis through numerical simulations.

## High Order Well-Balanced Discrete Kinetic Model for Shallow Water Equations

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In this work, we study a kinetic model that contains stiff relaxation terms in the source. This model can be applied to any non linear hyperbolic problem without source term, that we will call macroscopic problem, to obtain a larger system of equations with linear fluxes and non–linear source terms, the microscopic problem, that converges asymptotically to the original hyperbolic system.

## ADER and DeC: Arbitrarily High Order Explicit Methods for hyperbolic PDEs and ODEs

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The hyperbolic PDE community has used, in the last years, two powerful techniques that provide arbitrarily high order explicit schemes: ADER (arbitrary derivative) schemes and Deferred Correction (DeC) schemes.

## On modified Patankar schemes and oscillations: towards new stability definitions

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Modified Patankar (MP) schemes are linearly implicit ODE solvers for production destruction problems that guarantee unconditionally the positivity of the solutions and the conservation of the total quantities.

## Continuous Galerkin high order well-balanced discrete kinetic model for shallow water equations

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Kinetic models describe many physical phenomena, inter alia Boltzmann equations, but can also be used to approximate with an artificial relaxation procedure other macroscopic models. We consider the kinetic model proposed by Aregba-Driollet and Natalini, and we modify it in order to approximate shallow water (SW) equations. The difference with the original model stands in the presence of the source term in the SW equations due to the effect of the bathymetry. Thus, the kinetic model by Aregba-Driollet and Natalini must be extended in order to include this term and to maintain the asymptotic convergence to the macroscopic limit of the SW problem.

## Arbitrary High–Order Positivity–Preserving Finite–Volume Shallow–Water scheme without Restrictions on the CFL

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This is a talk is about a work in collaboration with Mirco Ciallella, Lorenzo Micalizzi and Philipp Öffner.

## Model order reduction for Friedrichs’ systems: a bridge between elliptic and hyperbolic problems

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Friedrichs’ systems (FS) K. O. Friedrichs. Comm. Pure & App. Math, 1958 are symmetric positive linear systems of first order PDEs that can describe many well known hyperbolic and elliptic problems in a unified framework. This allows, for example, to pass from one regime to another in different areas of the domain. One of the key ingredients of FS is the possibility of rewriting higher order derivative terms of PDEs through additional variables in the system of equations with only first order terms. This leads to a formulation composed by a linear combination of many block-structured fields $\mathcal{A}^k$ applied to the unknown $z$ and its first order derivatives, i.e., $\begin{cases} Az=f,\\ (\mathcal{D}-\mathcal{M}) z= 0, \end{cases} \qquad \text{ with } \qquad \begin{cases} Az = A_{(0)}z + A_{(1)}z,\\ A_{(0)}z = \mathcal{A}^0 z,\\ A_{(1)}z = \sum_{k=1}^d \mathcal{A}^k \partial_{x_k} z, \end{cases}$ where $\mathcal{D}$ and $\mathcal{M}$ are boundary fields, one given by the problem and the second used to impose the boundary conditions. Under some coercivity assumptions on the fields, the existence, uniqueness and well-posedness of the problem can be proven in different forms (strong, weak, ultraweak).

## A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity

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The Deferred Correction is an iterative procedure used to design numerical methods for systems of ODEs, characterized by an increasing accuracy at each iteration. The main advantage of this framework is the automatic way of getting arbitrarily high order methods, which can be put in Runge-Kutta form, based on the definition of subtimenodes in each timestep. The drawback is a larger computational cost with respect to the most used Runge-Kutta methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we remove the unnecessary subtimenodes in all the iterations, introducing interpolation processes between them. We provide the Butcher tableaux of the novel methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in the ODE and PDE settings.

## “Reduced Order Models on a Variational Multi-Scale Model of Navier–Stokes

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In this work, we study parametric incompressible flows given by Navier–Stokes equations. At the discrete level, we use continuous finite element (FE) method. The discrete model relies on a variational multi-scale (VMS) approach, which separates the large, sub-filter and small scales. The first two are resolved, while the last is not. The model includes sub-grid eddy viscosity to take care of the interaction between sub-filter and small scales. A local projection stabilization (LPS) term is introduced onto the sub-filter terms to provide stability. This term is based on interpolation and projection operators that penalizes the oscillations on the sub-filter scale. Moreover, in order to deal with no-slip boundary conditions without dramatically refining the mesh close to boundary layers, we resort to wall laws, which take into account the effect of the boundary already at a small distance from the boundary. The computational costs of such simulations, though being faster than other models as the small scales are not resolved, are still large, in particular in the time dependent case. Moreover, dealing with parametric problems in a multi-query context makes the computational burden unbearable. We propose a Galerkin projection of the equations onto a POD-generated reduced basis space as reduced order model (ROM). To take care of the nonlinearities of the problem, different hyper-reduction techniques are studied in order to obtain a reduced model that is independent of the dimension of the FE space. We provide simulations in two dimensions to validate the ROM and to prove the computational advantage of the approach.

## Model Order Reduction for Advection Dominated (Hyperbolic) Problems in an ALE framework

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Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov N-width of these problems. In the recent years, many new nonlinear algorithms and frameworks have been presented to overcome this issue. In this work, we propose a MOR technique for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. The key of the work consists of an arbitrary Lagrangian–Eulerian approach that modifies both the offline and online phases of the MOR process. This allows to calibrate the advected features on the same position and to strongly compress the reduced spaces. We will compare different MOR algorithms between the classical Greedy, EIM and POD and the more recent POD-NN. The calibration map is performed through an optimization process on a training set and then learned through polynomial regression and artificial neural networks for a quick evaluation in the online phase. In the performed simulations we show how the new algorithm defeats the classical method on many equations with nonlinear fluxes and with different boundary conditions. Finally, we compare the results obtained with different calibration maps.

## Global flux WENO finite volume and other structure preserving schemes for water equations

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Water wave equations are known to preserve various equilibria or some specific solutions. The lake at rest steady state is the most well-known equilibrium, and classically many methods are able to preserve such equilibrium. This allows capturing with extreme accuracy the perturbations of such a state.

## Saving computational costs with efficient iterative ADER methods: p-adaptivity, accuracy results and structure preserving limiters

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Hyperbolic solvers with arbitrarily high order of accuracy are widely used in scientific simulations, but they often come with a high computational cost. In this study, we introduce a modification to the ADER (Arbitrary DERivative) and Deferred Correction (DeC) methods that can save up to half of the computational cost, without sacrificing accuracy. By iteratively increasing the degree of solution reconstruction, our modification provides a natural framework for introducing p-adaptivity in the method, allowing users to adjust the accuracy level according to their goals, cell by cell. Additionally, our approach enables the preservation of solution properties such as positivity, local maximum principle or entropy inequalities, with a very efficient a posteriori limiter. We demonstrate the effectiveness of our method through results applied on ADER-DG and ADER-FV, using the Discrete Optimally increasing Order Method (DOOM) limiter to preserve positivity of density and pressure for compressible Euler and Navier-Stokes equations. Our approach offers a significant computational advantage compared to classical ADER methods, with minimal impact on the accuracy achieved.

## Arbitrary Lagrangian-Eulerian Model Reduction for Advection Dominated Problems and Some Graph Neural Network Ideas

Published:

Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov N-width of these problems. In the recent years, many new nonlinear algorithms and frameworks have been presented to overcome this issue. In this work, we propose a MOR technique for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. The key of the work consists of an arbitrary Lagrangian–Eulerian approach that modifies both the offline and online phases of the MOR process. This allows to calibrate the advected features on the same position and to strongly compress the reduced spaces. We will compare different MOR algorithms between the classical Greedy, EIM and POD and the more recent POD-NN. The calibration map is performed through an optimization process on a training set and then learned through polynomial regression and artificial neural networks for a quick evaluation in the online phase. In the performed simulations we show how the new algorithm defeats the classical method on many equations with nonlinear fluxes and with different boundary conditions. Finally, we compare the results obtained with different calibration maps. We extend the approach also for multiple tracking features and multiple dimensions.

## IMEX ADER and DeC: arbitrary high order schemes, stability and application to advection–diffusion–dispersion

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Arbitrary Derivative (ADER) [Dumbser et al., JCP, 2008] and Deferred Correction (DeC) [Abgrall, JSC, 2017] are arbitrarily high-order methods developed independently in distinct contexts, yet share notable similarities. Both methods employ an iterative process, incrementing by one the order of accuracy at each step. DeC originated as an ODE solver, then used also for more complicated space-time PDE discretizations, while ADER was initially explored as a PDE solver, particularly in its DG space-time discretization, but it has been investigated also as an ODE solver.

## Divergence-free Preserving Schemes: what’s wrong in SUPG and how to fix it

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The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.

## Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element

Published:

The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.

## Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element

Published:

The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.

## Taught course at University of Zurich

University Courses, Institute for Mathematics, University of Zurich, 2020

Teaching Assistant:

## High order accurate time integration methods

Doctoral school course, Ecole doctorale n°39, Université de Bordeaux, 2021

## High order accurate time integration methods

Doctoral school course, SISSA, 2023

## High order accurate time integration methods

Doctoral school course, SISSA, 2024