Uniform convergence of a linearly transformed particle method for the Vlasov-Poisson system

International audienceA particle method with linear transformation of the particle shape functions is studied for the 1d-1v Vlasov-Poisson equation, and a priori error estimates are proven which show that the approximated densities converge in the uniform norm. When compared to standard fixed-shape particle methods, the present approach can be seen as a way to gain one order in the convergence rate of the particle trajectories at the cost of linearly transforming each particle shape. It also allows to compute strongly convergent densities with particles that overlap in a bounded way

Bounded commuting projections for multipatch spaces with non-matching interfaces

We present commuting projection operators on de Rham sequences of two-dimensional multipatch spaces with local tensor-product parametrization and non-matching interfaces. Our construction yields projection operators which are local and stable in any $L^p$ norm with $p \in [1,\infty]$: it applies to shape-regular spline patches with different mappings and local refinements, under the assumption that neighboring patches have nested resolutions and that interior vertices are shared by exactly four patches. It also applies to de Rham sequences with homogeneous boundary conditions. Following a broken-FEEC approach, we first consider tensor-product commuting projections on the single-patch de Rham sequences, and modify the resulting patch-wise operators so as to enforce their conformity and commutation with the global derivatives, while preserving their projection and stability properties with constants independent of both the diameter and inner resolution of the patches

Convergence of a linearly transformed particle method for aggregation equations

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in $L^1 \cap L^p$ norm

Handling the divergence constraints in Maxwell and Vlasov-Maxwell simulations

International audienceThe aim of this paper is to review and classify the different methods that have been developed to enable stable long time simulations of the Vlasov-Maxwell equations and the Maxwell equations with sources. These methods can be classified in two types: field correction methods and sources correction methods. The field correction methods introduce new unknowns in the equations, for which additional boundary conditions are in some cases non trivial to find. The source correction consists in computing the sources so that they satisfy a discrete continuity equation compatible with a discrete Gauss' law that needs to be defined in accordance with the discretization of the Maxwell propagation operator

A broken FEEC framework for electromagnetic problems on mapped multipatch domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in [http://dx.doi.org/10.1090/mcom/3079, arXiv:2109.02553, our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge-Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy

Constructing exact sequences on non-conforming discrete spaces

AbstractIn this note, we propose a general procedure to construct exact sequences involving a non-conforming function space and we show how this construction can be used to derive a proper discrete Gauss law for structure-preserving discontinuous Galerkin (DG) approximations to the time-dependent 2d Maxwell equations

A δf\delta f PIC method with auxiliary Forward-Backward Lagrangian reconstructions

In this note we describe a $\delta f$ particle method where the bulk density is periodically remapped on a coarse spline grid using a Forward-Backward Lagrangian (FBL) approach. We describe the method in the case of an electrostatic PIC scheme and validate its qualitative properties using a classical two-stream instability subject to a uniform oscillating drive

Stable coupling of the Yee scheme with a linear current model

This work analyzes the stability of the Yee scheme for non stationary Maxwell's equations coupled with a linear current model. We show that the usual procedure may yield unstable scheme for physical situations that correspond to strongly magnetized plasmas in X-mode (TE) polarization. We propose to use first order clustered discretization of the vectorial product that gives back a stable coupling. We validate the schemes on some test cases representative of direct numerical simulations of X-mode in a magnetic fusion plasma