Victor Michel-Dansac

Giacomo Dimarco, Raphaël Loubère, Victor Michel-Dansac et al.

In this work, we consider the development of implicit explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The scheme proposed is asymptotically stable with a CFL condition independent from the Mach number and it degenerates in the low Mach number regime to a consistent discretization of the incompressible system. Since, it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) \cite{GotShuTad}, we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first to the second order both in space and time, which preserves the monotonicity of the solution, highly accurate for all choices of the Mach number and with a time step only restricted by the non stiff part of the system. In the last part, we show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties.

León Ávila, Manuel Castro, Victor Michel-Dansac et al.

This work presents a novel family of well-balanced numerical schemes for hyperbolic systems of balance laws based on the kinetic relaxation approach. The method begins by transforming the original non-linear system into a linearized kinetic system with an increased number of variables. In this framework, non-linearities are shifted to the source term, and the connection to the macroscopic variables is maintained via a projection operator related to the Maxwellian equilibrium states. These relaxed systems are typically solved using splitting techniques, where the evolution is decomposed into two distinct steps: first, the transport stage solving the source-free kinetic system, and second, the relaxation stage governed by an ordinary differential equation where the time derivative is balanced by the kinetic source term. The primary objective is to design schemes that accurately preserve the steady-state solutions of the original macroscopic equations, noting that these do not necessarily coincide with the trivial equilibria of the relaxed system. This well-balanced property is achieved through an appropriate formulation of the kinetic system combined with consistent reconstruction operators. This approach ensures that the balance is maintained throughout both the transport stage and the projection stage, where the treatment of the macroscopic source term is carefully integrated. We demonstrate that this framework can be implemented using both Finite Volume (FV) and Semi-Lagrangian (SL) techniques in the transport step. We develop and analyze these schemes for the Finite Volume case in both explicit and implicit formulations. Results are validated through numerical benchmarks, including the Burgers equation with a source term, 1D Shallow Water Equations (SWE) with bathymetry preserving non-trivial moving water equilibria and the Euler equations under external potentials.

Camilla Fiorini, Clément Flint, Louis Fostier et al.

Symbolic Regression (SR) is a widely studied field of research that aims to infer symbolic expressions from data. A popular approach for SR is the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework, which uses sparse regression to identify governing equations from data. This study introduces an enhanced method, Nested SINDy, that aims to increase the expressivity of the SINDy approach thanks to a nested structure. Indeed, traditional symbolic regression and system identification methods often fail with complex systems that cannot be easily described analytically. Nested SINDy builds on the SINDy framework by introducing additional layers before and after the core SINDy layer. This allows the method to identify symbolic representations for a wider range of systems, including those with compositions and products of functions. We demonstrate the ability of the Nested SINDy approach to accurately find symbolic expressions for simple systems, such as basic trigonometric functions, and sparse (false but accurate) analytical representations for more complex systems. Our results highlight Nested SINDy's potential as a tool for symbolic regression, surpassing the traditional SINDy approach in terms of expressivity. However, we also note the challenges in the optimization process for Nested SINDy and suggest future research directions, including the designing of a more robust methodology for the optimization process. This study proves that Nested SINDy can effectively discover symbolic representations of dynamical systems from data, offering new opportunities for understanding complex systems through data-driven methods.

Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri et al.

Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.