An adaptive numerical method for the Vlasov equation based on a multiresolution analysis

Besse, Nicolas; Filbet, Francis; Gutnic, Michaël; Paun, Ioana; Sonnendrücker, Éric

doi:10.1007/978-88-470-2089-4_41

Cited by 12 publications

(14 citation statements)

References 13 publications

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“…Therefore, according to the previous considerations, the Vlasov-Poisson periodic problem consists in finding a couple (f, E), smooth enough, periodic with respect to x, with period L, which solves the equations (2), (7), (8), and (9). Introducing the electrostatic potential φ = φ(t, x) such that E(t, x) = −∂ x φ(t, x), and denoting by G = G(x, y) the fundamental solution of the Laplacian operator in one dimension…”

Section: High-order Semi-lagrangian Schemes 95mentioning

confidence: 99%

“…But we cannot expect to obtain high-order estimates with this latter algorithm because we are restricted by the stability region phenomenon. Nevertheless, using the interpolatory wavelets of Deslaurier and Dubuc (wavelet built on Lagrange interpolation of any order) we could succeed in showing the convergence of the high-order adaptive schemes (see [9]). This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov--Poisson system

Besse

Mehrenberger

2008

Math. Comp.

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Abstract. In this paper we present some classes of high-order semi-Lagrangian schemes for solving the periodic one-dimensional Vlasov-Poisson system in phase-space on uniform grids. We prove that the distribution function f (t, x, v) and the electric field E(t, x) converge in the L 2 norm with a rate ofwhere m is the degree of the polynomial reconstruction, and ∆t and h are respectively the time and the phase-space discretization parameters.

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Section: High-order Semi-lagrangian Schemes 95mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov--Poisson system

Besse

Mehrenberger

2008

Math. Comp.

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“…Adaptive methods decrease computational cost drastically by keeping only a subset of all grid points. Some semi-Lagrangian adaptive methods have been developed, like in [10] and [2,7] where the authors use a moving grid or a multi-resolution analysis based on interpolating wavelets. But, few works use both approaches.…”

Section: Introductionmentioning

confidence: 99%

Load-Balancing for a Block-Based Parallel Adaptive 4D Vlasov Solver

Hoenen¹,

Violard²

2008

Lecture Notes in Computer Science

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Abstract. This work is devoted to the numerical resolution of the 4D Vlasov equation using an adaptive mesh of phase space. We previously proposed a parallel algorithm designed for distributed memory architectures. The underlying numerical scheme makes possible a parallelization using a block-based mesh partitioning. Efficiency of this algorithm relies on maintaining a good load balance during the whole simulation. In this paper, we propose a dynamic load balancing mechanism based on a relevant cost metric and a geometric partitioning algorithm. This mechanism is deeply integrated into the parallel algorithm in order to minimize overhead. Performance measurements on a PC cluster show the good quality of our load balancing and confirm the pertinence of our approach.

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“…In order to save computational resources while approximating the complex and thin structures that may appear in the solutions of the Vlasov equation, several adaptive schemes have been proposed in the past few years, see in particular (Besse et al, 2001;Gutnic et al, 2004;Sonnendrücker et al, 2004), where the authors use moving phase-space grids or interpolatory wavelets. Originated in the semi-Lagrangian method of (Cheng and Knorr, 1976), later revisited by Sonnendrücker et al (1999), these schemes gave encouraging results in practice, but none was proven to be more efficient than the uniform ones.…”

Section: Introductionmentioning

confidence: 99%

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

Pinto¹

2007

International Journal of Applied Mathematics and Computer Science

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This article aims at giving a simplified presentation of a new adaptive semi-Lagrangian scheme for solving the (1 + 1)-dimensional Vlasov-Poisson system, which was developed in 2005 with Michel Mehrenberger and first described in (Campos Pinto and Mehrenberger, 2007). The main steps of the analysis are also given, which yield the first error estimate for an adaptive scheme in the context of the Vlasov equation. This article focuses on a key feature of our method, which is a new algorithm to transport multiscale meshes along a smooth flow, in a way that can be said optimal in the sense that it satisfies both accuracy and complexity estimates which are likely to lead to optimal convergence rates for the whole numerical scheme. From the regularity analysis of the numerical solution and how it gets transported by the numerical flow, it is shown that the accuracy of our scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions are proved to converge in L ∞ towards the exact ones as ε and Δt tend to zero, and in addition to the numerical tests presented in (Campos Pinto and Mehrenberger, 2007), some complexity bounds are established which are likely to prove the optimality of the meshes.

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An adaptive numerical method for the Vlasov equation based on a multiresolution analysis

Cited by 12 publications

References 13 publications

Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov--Poisson system

Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov--Poisson system

Load-Balancing for a Block-Based Parallel Adaptive 4D Vlasov Solver

A Direct and Accurate Adaptive Semi-Lagrangian Scheme for the Vlasov-Poisson Equation

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