Asymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling

In this work, we develop a new class of numerical schemes for collisional kinetic equations in the diffusive regime. The first step consists in reformulating the problem by decomposing the solution in the time evolution of an equilibrium state plus a perturbation. Then, the scheme combines a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part, and is designed in such a way to be uniformly stable with respect to the diffusive scaling and to be consistent with the asymptotic diffusion equation. Moreover, since particles are only used to describe the perturbation part of the solution, the scheme becomes computationally less expensive – and is thus an asymptotically complexity diminishing scheme (ACDS) – as the solution approaches the equilibrium state due to the fact that the number of particles diminishes accordingly. This contrasts with standard methods for kinetic equations where the computational cost increases (or at least does not decrease) with the number of interactions. At the same time, the statistical error due to the Monte Carlo part of the solution decreases as the system approaches the equilibrium state: the method automatically degenerates to a solution of the macroscopic diffusion equation in the limit of infinite number of interactions. After a detailed description of the method, we perform several numerical tests and compare this new approach with classical numerical methods on various problems up to the full three dimensional case

The Moment Guided Monte Carlo method for the Boltzmann equation

In this work we propose a generalization of the Moment Guided Monte Carlo method developed in [11]. This approach permits to reduce the variance of the particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a non equilibrium term. Here, at the contrary to the previous work in which we considered the simplified BGK operator, we deal with the full Boltzmann operator. Moreover, we introduce an hybrid setting which permits to entirely remove the resolution of the kinetic equation in the limit of infinite number of collisions and to consider only the solution of the compressible Euler equation. This modification additionally reduce the statistical error with respect to our previous work and permits to perform simulations of non equilibrium gases using only a few number of particles. We show at the end of the paper several numerical tests which prove the efficiency and the low level of numerical noise of the method.Comment: arXiv admin note: text overlap with arXiv:0908.026

A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit

We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments.Comment: 20 pages, 7 figures, published in Monte Carlo and Quasi-Monte Carlo Methods 2018, correction of minor typographical error

Numerical resolution of conservation laws with OpenCL

We present several numerical simulations of conservation laws on recent multicore processors, such as GPU’s, using the OpenCL programming framework. Depending on the chosen numerical method, different implementation strategies have to be considered, for achieving the best performance. We explain how to program efficiently three methods: a finite volume approach on a structured grid, a high order Discontinuous Galerkin (DG) method on an unstructured grid and a Particle-In-Cell (PIC) method. The three methods are respectively applied to a two-fluid computation, a Maxwell simulation and a Vlasov-Maxwell simulation.

An axisymmetric PIC code based on isogeometric analysis

Isogeometric analysis has been developed recently to use basis functions resulting from the CAO description of the computational domain for the finite element spaces. The goal of this study is to develop an axisymmetric Finite Element PIC code in which specific spline Finite Elements are used to solve the Maxwell equations and the same spline functions serve as shape function for the particles. The computational domain itself is defined using splines or NURBS

A HYBRID METHOD FOR ANISOTROPIC ELLIPTIC PROBLEMS BASED ON THE COUPLING OF AN ASYMPTOTIC-PRESERVINGMETHOD WITH THE ASYMPTOTIC LIMIT MODEL

International audienceThis paper presents a hybrid numerical method to solve efficiently a class of highly anisotropic elliptic problems. The anisotropy is aligned with one coordinate axis and its strength is described by a parameter ε ∈ (0, 1], which can largely vary in the study domain. Our hybrid model is based on asymptotic techniques and couples (spatially) an asymptotic-preserving model with its asymptotic limit model, the latter being used in regions where the anisotropy parameter ε is small. Adequate coupling conditions link the two models. The aim of this hybrid procedure is to reduce the computational time for problems where the region of small ε-values extends over a significant part of the domain, and this is due to the reduced complexity of the limit model