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The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved.

In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.

The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.

We analyze randomized matrix-free quadrature algorithms for spectrum and spectral sum approximation. The algorithms studied include the kernel polynomial method and stochastic Lanczos quadrature, two widely used methods for these tasks. Our analysis of spectrum approximation unifies and simplifies several one-off analyses for these algorithms which have appeared over the past decade. In addition, we derive bounds for spectral sum approximation which guarantee that, with high probability, the algorithms are simultaneously accurate on all bounded analytic functions. Finally, we provide comprehensive and complimentary numerical examples. These examples illustrate some of the qualitative similarities and differences between the algorithms, as well as relative drawbacks and benefits to their use on different types of problems.

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.

An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.

We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error analysis in [9] for Runge-Kutta methods for nonlinear parabolic equations; in analogy to [9], the proofs are based on maximal regularity properties of discontinuous Galerkin methods for non-autonomous linear parabolic equations.

We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie--Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin's model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie--Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.