arXiv subjects

Giuseppe Izzo

Publications and source records attributed to Giuseppe Izzo.

4 recordsLinked to original sources

arXiv2026

High Order semi-implicit Rosenbrock type and Multistep methods for evolutionary partial differential equations with higher order derivatives

The aim of this work is to apply a semi-implicit (SI) strategy within a Rosenbrock-type and IMEX linear multistep (LM) framework to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy provides great flexibility to treat these equations, and allows the construction of simple lienarly implicit schemes without any Newton iteration. Furthermore, the SI schemes so designed do not require the severe time-step restrictions typically encountered when using explicit methods for stability, i.e., $\Delta t = \mathcal{O}(\Delta x^k)$ for the $k$-th order PDEs with $k\ge 2$. For space discrertization, this strategy is combined with finite difference schemes. We provide example of methods up to order $p = 4$, and we illustrate the effectiveness of the schemes with appllications to dissipative, dispersive, and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and achieve the expected orders of accuracy.

math.NA↗

arXiv2024

Modified Patankar Linear Multistep methods for production-destruction systems

Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weight denominators to achieve it.

math.NA↗

arXiv2022

The effect of time discretization on the solution of parabolic PDEs with ANNs

We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $θ$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties of the methods. The results of numerical experiments confirm that ELM-based solution techniques combined with BDF methods can provide high-accuracy solutions of parabolic PDEs.

math.NA↗

arXiv2018

Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part

For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit-explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by implicit formula, and the non-stiff part is integrated by an explicit formula. We will construct methods where the explicit part has strong stability preserving (SSP) property, and the implicit part of the method is $A$-, or $L$-stable. We will also investigate stability of these methods when the implicit and explicit parts interact with each other. To be more precise, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is $A$-, or $L$-stable. Finally we furnish examples of SSP IMEX DIMSIMs up to the order four with good stability properties.

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