# 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 a postdoctoral researcher in Numerical Analysis, passionate about Data and musician in spare time.

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

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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

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

## Weighted Model Order Reduction to Quantify Uncertainties

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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 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.

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

Published in arXiv preprint, 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. (2022). "A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity. " arXiv preprint, 2022. https://arxiv.org/abs/2210.02976. https://arxiv.org/abs/2206.03889
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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 arXiv preprint, 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." (2022) arXiv preprint, arXiv:2211.14528. https://arxiv.org/abs/2211.14528
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## Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order

Published in arXiv preprint, 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." (2022) arXiv preprint, arXiv:2212.07783. https://arxiv.org/abs/2212.07783
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## ADER and DeC: Arbitrarily High Order Explicit Time Integration Methods

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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

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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.

## High order accurate time integration methods

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