arXiv subjects

Jean-Christophe Nave

Publications and source records attributed to Jean-Christophe Nave.

At least 19 recordsLinked to original sources

arXiv2026

A Hybrid semi-Lagrangian Flow Mapping Approach for Vlasov Systems: Combining Iterative and Compositional Flow Maps

We propose a hybrid semi-Lagrangian scheme for the Vlasov--Poisson equation that combines the Numerical Flow Iteration (NuFI) method with the Characteristic Mapping Method (CMM). Both approaches exploit the semi-group property of the underlying diffeomorphic flow, enabling the reconstruction of solutions through flow maps that trace characteristics back to their initial positions. NuFI builds this flow map iteratively, preserving symplectic structure and conserving invariants, but its computational cost scales quadratically with time. Its advantage lies in a compact, low-dimensional representation depending only on the electric field. In contrast, CMM achieves low computational costs when remapping by composing the global flow map from explicitly stored submaps. The proposed hybrid method merges these strengths: NuFi is employed for accurate and conservative local time stepping, while CMM efficiently propagates the solution through submap composition. This approach reduces storage requirements, maintains accuracy, and improves structural properties. Numerical experiments demonstrate the effectiveness of the scheme and highlight the trade-offs between memory usage and computational cost. We benchmark against a semi-Lagrangian predictor-corrector scheme used in modern gyrokinetic codes, evaluating accuracy and conservation properties.

math.NA↗

arXiv2025

Noncommutative Laplacian and numerical approximation of Laplace-Beltrami spectrum of compact Riemann surfaces

We derive a numerical approximation of the Laplace-Beltrami operator on compact surfaces embedded in $\mathbb{R}^3$ with an axial symmetry. To do so we use a noncommutative Laplace operator defined on the space of finite dimensional hermitian matrices. This operator is derived from a foliation of the surface obtained under an $S^1$-action on the surface. We present numerical results in the case of the sphere and a generic ellipsoid.

math.NA↗

arXiv2025

Structure preserving discretization: A Berezin-Toeplitz Quantization viewpoint

In this paper, we introduce a comprehensive axiomatization of structure-preserving discretization through the framework of commutative diagrams. By establishing a formal language that captures the essential properties of discretization processes, we provide a rigorous foundation for analyzing how various structures (such as algebraic, geometric, and topological features) are maintained during the transition from continuous to discrete settings. Specifically, we establish that the transition from continuous to discrete differential settings invariably leads to noncommutative structures, reinforcing previous observation on the interplay between discretization and noncommutativity. We demonstrate the applicability of our axiomatization by applying it to the Berezin-Toeplitz quantization, showing that this quantization method adheres to our proposed criteria for structure-preserving discretization. We establish in this setting a precise limit theorem for the approximation of the Laplacian by a sequence of matrix approximations. This work not only enriches the theoretical understanding of the nature of discretization but also sets the stage for further exploration of its applications across various discretization methods.

math.NA↗

arXiv2025

Diffeomorphic Neural Operator Learning

We present an operator learning approach for a class of evolution operators using a composition of a learned lift into the space of diffeomorphisms of the domain and the group action on the field space. In turn, this transforms the semigroup structure of the evolution operator into a corresponding group structure allowing time stepping be performed through composition on the space of diffeomorphisms rather than in the field space directly. This results in a number of structure-preserving properties related to preserving a relabelling symmetry of the dynamics as a hard constraint. We study the resolution properties of our approach, along with its connection to the techniques of diffeomorphic image registration. Numerical experiments on forecasting turbulent fluid dynamics are provided, demonstrating its conservative properties, non-diffusivity, and ability to capture anticipated statistical scaling relations at sub-grid scales. Our method provides an example of geometric operator learning and indicates a clear performance benefit from leveraging a priori known infinite-dimensional geometric structure.

math.NA↗

arXiv2024

A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.

math.NA↗

arXiv2024

Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence

In this paper, we present a methodology for establishing constructive proofs of existence of smooth, stationary, non-radial localized patterns in the planar Swift-Hohenberg equation. Specifically, given an approximate solution $u_0$, we construct an approximate inverse for the linearization around $u_0$, enabling the development of a Newton-Kantorovich approach. Consequently, we derive a sufficient condition for the existence of a unique localized pattern in the vicinity of $u_0$. The verification of this condition is facilitated through a combination of analytic techniques and rigorous numerical computations. Moreover, an additional condition is derived, establishing that the localized pattern serves as the limit of a family of periodic solutions (in space) as the period tends to infinity. The integration of analytical tools and meticulous numerical analysis ensures a comprehensive validation of this condition. To illustrate the efficacy of the proposed methodology, we present computer-assisted proofs for the existence of three distinct unbounded branches of periodic solutions in the planar Swift-Hohenberg equation, all converging towards a localized planar pattern, whose existence is also proven constructively. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \cite{julia_cadiot}.

math.AP↗

arXiv2024

A Characteristic Mapping Method for Vlasov-Poisson with Extreme Resolution Properties

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method. The extreme fine-scale resolution features are illustrated showing the method's capabilities to efficiently treat filamentation in fusion plasma simulations.

math.NA↗

arXiv2024

Viscous shock waves of Burgers equation with fast diffusion and singularity

In this paper, we study the asymptotic stability of viscous shock waves for Burgers' equation with fast diffusion $u_t+f(u)_x=μ(u^m)_{xx}$ on $\mathbb{R} \times (0, +\infty)$ when $0 u_+=0$, the equation with fast diffusion $(u^m)_{xx}=m\left(\frac{u_x}{u^{1-m}}\right)_x$ processes a strong singularity at $u_+=0$, which causes the stability study to be challenging. We observe that, there exist two different types of viscous shocks, one is the non-degenerate shock satisfying Lax's entropy condition with fast algebraic decay to the singular state $u_+=0$, which causes much strong singularity to the system in the form of $m\left(\frac{u_x}{u^{1-m}}\right)_x$, and the other is the degenerate viscous shock with slow algebraic decay to $u_+=0$, which makes less strong singularity to the system. In order to overcome the singularity at $u_+=0$, we technically use the weighted energy method and develop a new strategy where the weights related to the shock waves are carefully selected, while the chosen weights for the non-degenerate case are stronger than the degenerate case. Numerical simulations are also carried out in different cases to illustrate and validate our theoretical results. In particular, we numerically approximate the solution for different value of $0<m<1$, and find that the shapes of shock waves become steeper when the singularity $\left(\frac{u_x}{u^{1-m}}\right)_x$ is stronger as $m\rightarrow 0$, which indicates that the effect of singular fast diffusion on the solution is essential.

math.AP↗

arXiv2024

Singularity formation of vortex sheets in 2D Euler equations using the characteristic mapping method

The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for non-smooth initial data, in particular for vortex sheets. To this end high resolution computations of vortex layers in 2D incompressible Euler flows are performed using the characteristic mapping method (CMM). This semi-Lagrangian method evolves the flow map using the gradient-augmented level set method (GALS). The semi-group structure of the flow map allows its decomposition into sub-maps (each over a finite time interval), and thus the precision can be controlled by choosing appropriate remapping times. Composing the flow map yields exponential resolution in linear time, a unique feature of CMM, and thus fine scale flow structures can be resolved in great detail. Here the roll-up process of vortex layers is studied varying the thickness of the layer showing its impact on the growth of palinstrophy and possible blow up of absolute vorticity. The curvature of the vortex sheet shows a singular-like behavior. The self-similar structure of the vortex core is investigated in the vanishing thickness limit. Conclusions on the non-uniqueness of weak solutions of 2D Euler for non-smooth initial data are drawn and the presence of flow singularities is revealed tracking them in the complex plane.

physics.flu-dyn↗

arXiv2024

Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.

math.AP↗

arXiv2024

Nonlinear stability of shock profiles to Burgers' equation with critical fast diffusion and singularity

In this paper we propose the first framework to study Burgers' equation featuring critical fast diffusion in form of $u_t+f(u)_x = (\ln u)_{xx}$. The solution possesses a strong singularity when $u=0$ hence bringing technical challenges. The main purpose of this paper is to investigate the asymptotic stability of viscous shocks, particularly those with shock profiles vanishing at the far field $x=+\infty$. To overcome the singularity, we introduce some weight functions and show the nonlinear stability of shock profiles through the weighted energy method. Numerical simulations are also carried out in different cases of fast diffusion with singularity, which illustrate and confirm our theoretical results.

math.AP↗

arXiv2024

Emergence of Collective Behaviors from Local Voronoi Topological Perception

This article addresses how diverse collective behaviors arise from simple and realistic decisions made entirely at the level of each agent's personal space in the sense of the Voronoi diagram. We present a discrete time model in 2D in which individual agents are aware of their local Voronoi environment and may seek static target locations. In particular, agents only communicate directly with their Voronoi neighbors and make decisions based on the geometry of their own Voronoi cells. With two effective control parameters, it is shown numerically to capture a wide range of collective behaviors in different scenarios. Further, we show that the Voronoi topology facilitates the computation of several novel observables for quantifying discrete collective behaviors. These observables are applicable to all agent-based models and to empirical data.

physics.soc-ph↗

arXiv2023

A characteristic mapping method for incompressible hydrodynamics on a rotating sphere

We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as a composition of sub-interval flows formed by $C^1$ spherical spline interpolants. This approximation technique has the capacity of resolving sub-grid scales generated over time without increasing the spatial resolution of the computational grid. The numerical method is analyzed and validated using standard test cases yielding third-order accuracy in the supremum norm. Numerical experiments illustrating the unique resolution properties of the method are performed and demonstrate the ability to reproduce the forward energy cascade at sub-grid scales by upsampling the numerical solution.

math.NA↗

arXiv2023

Statistical Fluctuation of Infinitesimal Spaces

This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the definition of finite dimensional compatible Dirac operators using Clifford algebras. This definition also leads to a new construction of a Laplace operator. On the other hand, after a well-chosen Green's function defined on a manifold, we show that when the Dirac operators can be interpreted as stochastic matrices. The sequence $(D_n)_{n\in \mathbb{N}}$ converges then in average to the usual Dirac operator on a spin manifold. The same conclusion can be drawn for the Laplace operator.

math.NA↗

arXiv2023

Noncommutative Differential Geometry on Infinitesimal Spaces

In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a $C^*$-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function $C(M)$ over a manifold $M$ can be approximated by a direct limit of $C^*$-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the $d$-lattice in $\mathbb{R}^d$ and for the torus $\mathbb{T}^d$.

math.NA↗

arXiv2023

A Characteristic Mapping Method for Tracer Transport on the Sphere

A semi-Lagrangian Characteristic Mapping method for the solution of the tracer transport equations on the sphere is presented. The method solves for the solution operator of the equations by approximating the inverse of the diffeomorphism generated by a given velocity field. The evolution of any tracer and mass density can then be computed via pullback with this map. We present a spatial discretization of the manifold-valued map using a projection-based approach with spherical spline interpolation. The numerical scheme yields $C^1$ continuity for the map and global second-order accuracy for the solution of the tracer transport equations. Error estimates are provided and supported by convergence tests involving solid body rotation, moving vortices, deformational, and compressible flows. Additionally, we illustrate some features of computing the solution operator using a numerical mixing test and the transport of a fractal set in a complex flow environment.

math.NA↗

arXiv2022

High-order FDTD schemes for Maxwell's interface problems with discontinuous coefficients and complex interfaces based on the Correction Function Method

We propose high-order FDTD schemes based on the Correction Function Method (CFM) for Maxwell's interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an interface to retain the order of a finite difference approximation. For this, we solve a system of PDEs based on the original problem by minimizing an energy functional. The CFM is applied to the standard Yee scheme and a fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D using the transverse magnetic mode (TM$_z$). Numerical examples include scattering of magnetic and non-magnetic dielectric cylinders, and problems with manufactured solutions using various complex interfaces and discontinuous piecewise varying coefficients. Long-time simulations are also performed to provide numerical evidences of the stability of the proposed numerical approach. The proposed CFM-FDTD schemes achieve up to fourth-order convergence in $L^2$-norm and provide approximations devoid of spurious oscillations.

math.NA↗

arXiv2021

Conservative Integrators for Vortex Blob Methods

Conservative symmetric second-order one-step integrators are derived using the Discrete Multiplier Method for a family of vortex-blob models approximating the incompressible Euler's equations on the plane. Conservative properties and second order convergence are proved. A rational function approximation was used to approximate the exponential integral that appears in the Hamiltonian. Numerical experiments are shown to verify the conservative property of these integrators, their second-order accuracy, and as well as the resulting spatial and temporal accuracy of the vortex blob method. Moreover, the derived implicit conservative integrators are shown to be better at preserving conserved quantities than standard higher-order explicit integrators on comparable computation times.

math.NA↗