Abstract In recent years, the efficient numerical solution of Hamiltonian problems has been tackled by means of Hamiltonian Boundary Value Methods (HBVMs) (see, e.g., the monograph [1]). Such methods can be defined through a truncated expansion of the vector field defining the problem, $$ \dot y(ch) = f(y(ch)), \qquad c\in[0,1], \qquad y(0)=y_0\in\mathbb{R}^m, $$ along the Legendre polynomial basis orthonormal on $[0,1]$, $\int_0^1P_i(c)P_j(c)\mathrm{d} c = \delta_{ij}$, $i,j=0,1,\dots$. For a given $s\in\mathbb{N}$, this results into a polynomial vector field $\dot\sigma(ch)$, of degree $s-1$, defined as: $$ \begin{array}{l} \dot \sigma(ch) = \sum_{j=0}^{s-1} P_j(c) \gamma_j(\sigma), \quad c\in[0,1], \quad \sigma(0)=y_0\in\mathbb{R}^m, \\[8pt] \gamma_j(\sigma)=\int_0^1P_j(\tau)f(\sigma(\tau h))\mathrm{d}\tau. \end{array} $$ As is clear [1,2], this is equivalent to require the residual be orthogonal to all polynomials of degree $s-1$, w.r.t. the $L_2[0,1]$ product: $$\int_0^1 P_j(\tau)\left[\dot\sigma(\tau h)-f(\sigma(\tau h))\right]\mathrm{d} \tau = 0, \qquad j=0,\dots,s-1.$$ Discretizing the Fourier coefficients $\gamma_j(\sigma)$ by means of a Gauss-Legendre quadrature of order $2k$, $k\ge s$, results into a HBVM$(k,s)$ method. Though formally a $k$-stage Runge-Kutta method [1], the corresponding discrete problem can be always recast in terms of the $s$ Fourier coefficients, independently of $k$, so that the accurate approximation of the inetgrals is not an issue. Moreover, the availability of a very efficient nonlinear blended iteration for solving the generated discrete problems [1,3], allows using large values of $s$ coupled with huge timesteps $h$, resulting into spectrally accurate in time methods [4]. These latter turn out to be particularly effective when solving large-size semi-linear ODE problems deriving from the space semi-discretization of Hamiltonian PDEs [5]. This approach has been recently extended to cope with the efficient numerical solution of Fractional Differential Equations, which we take in the form, for $\alpha\in(0,1)$, $$ y^{(\alpha)}(ch) = f(y(ch)), \qquad c\in[0,1], \qquad y(0)=y_0\in\mathbb{R}^m, $$ with $y^{(\alpha)}(t) = \frac{1}{\Gamma(1-\alpha)}\int_0^t(t-x)^{-\alpha}y'(x)\mathrm{d} x$~ the Caputo fractional derivative. In such a case, a polynomial vector field, defining a corresponding Fractional HBVM (FHBVM), is derived by considering a truncated expansion of the vector field along an orthonormal Jacobi basis naturally induced by the problem, $\int_0^1\omega(c)P_i(c)P_j(c)\mathrm{d} c = \delta_{ij}$, ~$i,j=0,1,\dots$,~ $\omega(c)=\alpha(1-c)^{\alpha-1}$: $$ \begin{array}{l} \sigma^{(\alpha)}(ch) = \sum_{j=0}^{s-1}P_j(c)\gamma_j(\sigma), \quad c\in[0,1], \quad \sigma(0)=y_0\in\mathbb{R}^m, \\[8pt] \gamma_j(\sigma)=\int_0^1\omega(\tau)P_j(\tau)f(\sigma(\tau h))\mathrm{d}\tau. \end{array} $$ Again, this approximated polynomial vector field is equivalent to requiring that the residual be orthogonal to all polynomials of degree $s-1$, w.r.t. the corresponding weighted $L_2[0,1]$ product: $$\int_0^1 \omega(\tau)P_j(\tau)\left[\sigma^{(\alpha)}(\tau h)-f(\sigma(\tau h))\right]\mathrm{d} \tau = 0, \qquad j=0,\dots,s-1.$$ Discretization of the Fourier coefficients $\gamma_j(\sigma)$ by using a Gauss-Jacobi quadrature of order $2k$, $k\ge s$, then results into a FHBVM$(k,s)$ method (as is clear, this reduces to a HBVM$(k,s)$ method, when $\alpha=1$) [6]. Also in this case, the availability of an efficient nonlinear iteration, allowing using large values of $s$, coupled when needed with a graded mesh to cope with possible nonsmooth vector fields, allows reaching spectrally accurate in time solutions. FHBVMs have been implemented into two Matlab codes, {\tt fhbvm} [7] and {\tt fhbvm2} [8]. A systematic comparison on the set of test problems collected in the FDE-Testset [9] clearly shows the effectiveness of such codes, w.r.t. existing Matlab codes available on the web.

Abstract Autoresonance is a phenomenon of physical interest that takes place when a nonlinear oscillator automatically adjusts its instantaneous frequency so as to match the varying frequency of a forcing term, causing that the amplitude of the oscillations keeps growing. It may happen even if the frequency of the forcing term varies slowly. The stroboscopic averaging method (SAM), which provides an efficient numerical technique for the integration of highly oscillatory systems, cannot be used directly to study autoresonance due to the slow changes of the forcing frequency. We study how to modify SAM to cater for such slow variations and include numerical experiments for the Duffing oscillator showing the computational advantages of using SAM.

Abstract To be announced

Abstract Standard boundary integral equation techniques do not naturally result in 2nd-kind integral equations for magnetostatics in the presence of an open surface. We begin by reviewing the classical formulation for closed surfaces and numerical solvers for axisymmetric boundaries. We then formulate a second-kind integral equation for the genus-zero open-surface case and introduce a consistency condition with augmented representation of the fields that ensures well-posedness when the boundary genus exceeds zero. Finally, we explore scattering from several open surfaces using the method of reflections. We will take a look at the numerical difficulties and future work that arise from these problems.

Abstract We investigate the control and coordination of artificial microswimmers inspired by breaststroke-like propulsion within a geometric framework. At low Reynolds numbers, locomotion arises from non-reciprocal shape changes, leading naturally to a formulation in terms of sub-Riemannian geometry. We first analyze the configuration dynamics of a single swimmer. Starting from a three-arm model and reducing to a two-arm subsystem, we show that the resulting control system induces a rank-two distribution with Lie algebra growth corresponding to a Cartan $(2,3,5)$ structure. This characterization yields controllability, highlights the role of abnormal extremals, and connects the model to classical articulated swimmers such as the Purcell three-link swimmer. Numerical simulations evaluate propulsion efficiency under different criteria. Motivated by the need for collective effects at microscopic scales, we study decentralized coordination strategies for swarms of microswimmers. Each agent adjusts its shape variables using only local information exchanged with neighboring agents. The interaction structure is modeled by a fixed communication graph, and the resulting feedback law is formulated using the associated graph Laplacian. We provide convergence estimates for the synchronization dynamics and illustrate the behavior of the swarm through numerical simulations under various interaction scenarios.

Abstract In recent years, the development and analysis of reaction-diffusion models have gained increasing attention due to their applicability in various sustainability-related contexts. Notable examples include the modeling of ecological dynamics, such as vegetation patterns in semi-arid ecosystems [1]. A further relevant application, in the framework of sustainable innovation, arises in the mathematical modeling of supply chains and production networks, where continuum descriptions based on Partial Differential Equations (PDEs) can be used to represent the evolution of material and product flows across interconnected processors, buffers and distribution nodes [6]. These models are generally described by systems of PDEs that, after space discretization, typically lead to high-dimensional stiff systems of ordinary differential equations, whose efficient and reliable time integration remains a challenging task. In this talk, we discuss recent developments in Runge--Kutta (RK) methods designed for the numerical solution of such stiff problems. Starting from the framework of Time-Accurate and highly Stable Explicit RK (TASE--RK) methods [3, 4], we present adapted strategies for reaction--diffusion systems. In particular, suitable operators are used to improve the stability properties of explicit schemes without compromising computational efficiency. The resulting General TASE--RK methods are designed to reduce error constants and computational effort for the class of problems under consideration, while getting suitable stability properties such as strong $A(\theta)$-stability [5]. Furthermore, we focus on the derivation of a more flexible class of linearly implicit schemes, referred to as Modified TASE--RK methods \cite{AcetoContePagano2025}. In this setting, different operators may be employed at the internal stages, allowing to move beyond the standard TASE--RK formulation. By exploiting the relation with W-methods, schemes of order up to four are derived with a reduced number of linear systems with respect to classical TASE--RK methods. The proposed strategy also permits the design of strongly $A$-stable methods, which are suitable for advection--reaction--diffusion models. The theoretical properties of the presented methods are supported by several numerical experiments.

Abstract We extend the classical stability analysis of disease-free equilibria in epidemiological compartmental problems. In particular, we construct a method to examine whether disease-free orbits attract nearby orbits (disease-free or not). In the theory of compartmental epidemic dynamical models, a clear distinction is made between infected compartments and disease-free compartments. Typically, the behavior of the system is studied in the neighborhood of a disease-free equilibrium. Instead, we subdivide the space of disease-free states into regions where they attract nearby orbits and regions where they do not attract them. More precisely, we subdivide the space of disease-free states into regions where either (a) all eigenvalues of the Jacobian of the system have negative real parts, or (b) at least one eigenvalue has a positive real part, or (c), the border case: not in (a) or (b). In general, this is a difficult task. But extensive numerical tests and graphical illustrations show the practical applicability of our method in understanding the topology of the phase portraits and their global behavior. We consider some classically important models and hypothesize that the same results hold for a large class of epidemic models.

Abstract In recent years, the development of efficient numerical methods for large-scale reaction-diffusion systems has become increasingly important in several sustainability-oriented industrial applications. In particular, predictive models for material degradation and corrosion processes play a key role in improving the reliability and lifetime of engineering infrastructures, supporting maintenance strategies, and reducing disruptions in production and supply-chain systems. The mathematical models arising in these contexts are often governed by partial differential equations [4]. In this talk, we focus on the simulation of corrosion processes through a phase-field model for metal pitting corrosion [5]. Phase-field approaches are especially attractive because they naturally describe the evolution of corrosion fronts and complex moving interfaces without requiring costly remeshing techniques. Nevertheless, the resulting equations are highly stiff, making standard explicit time integrators inefficient due to severe stability constraints [2, 3]. To overcome these issues, we propose a matrix-oriented strategy that exploits the Kronecker sum structure of the discrete Laplacian operator arising from finite difference discretizations on rectangular domains. This allows the construction of efficient schemes that avoid large matrix inversions while remaining applicable also to non-rectangular geometries [1]. The proposed methods are assessed through numerical experiments in two- and three-dimensional settings and compared with existing approaches. The results show significant improvements in computational efficiency together with accurate resolution of the corrosion dynamics, highlighting the potential of the proposed methodology for fast and reliable corrosion simulations in realistic engineering applications [1].

Abstract Fourth order space semi-discretizations for advection diffusion reaction PDEs with a marked parabolic character have some practical importance, since they significantly reduce the stiffness and the computational costs of the resulting ODE systems when compared with standard second order space discretizations. Here, we consider a very simple fourth order scheme based on central finite differences for the case of Boundary conditions of Dirichlet type and an initial condition on hyper-rectangular domains. The main goal of the talk is to show the progress in the research about the fourth order of convergence, which is also kept for the $m$-dimensional case.