On streamline diffusion schemes for the one and one-half dimensional relativistic Vlasov–Maxwell system

Standar, Christoffer

doi:10.1007/s10092-015-0141-4

Cited by 3 publications

(7 citation statements)

References 9 publications

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“…We performed the calculations for the simplified case of one space variable and two velocities variables, i.e. the one and one-half dimensional Vlasov-Maxwell system (cf [22]), which takes the following form:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Numerical approaches for the VM system have been considered by several authors in different setting. The most relevant studies to this work are given by Gamba and co-workers [9] devoted to a discontinuous Galerkin approach, and Standar in [22] where the stability and a priori error estimates for the h version of SD method for VM are derived. As some related studies we mention the analysis of a one dimensional model problem for the relativistic VM system in an interval given by Filbet and co-workers in [13].…”

Section: Introductionmentioning

confidence: 99%

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On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

Asadzadeh¹,

Kowalczyk²,

Standar³

2019

Kinetic &Amp; Related Models

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We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space H s+1 (Ω), we derive global a priori error bound of order O(h/p) s+1/2 , where h(= max K h K ) is the mesh parameter and p(= max K p K ) is the spectral order. This estimate is based on the local version with h K = diam K being the diameter of the phase-space-time element K and p K is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h 2 + k 2 ), where h is the spatial mesh size and k is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions.Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [20].

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

Asadzadeh¹,

Kowalczyk²,

Standar³

2019

Kinetic &Amp; Related Models

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“…We present in this section results of some numerical implementations that justify the accuracy of the SD method. The computations are performed for the simplified case of one spatial variable and two velocities (cf., Standar 2016), which takes the following form:…”

Section: Numerical Resultsmentioning

confidence: 99%

Convergence of hp-Streamline Diffusion Method for Vlasov–Maxwell System

Asadzadeh

Kowalczyk

Standar

2019

Journal of Computational and Theoretical Transport

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In this paper we study stability and convergence for hp-streamline diffusion (SD) finite element method for the, relativistic, timedependent Vlasov-Maxwell (VM) system. We consider spatial domain X x & R 3 and velocities v 2 X v & R 3 : The objective is to show globally optimal a priori error bound of order Oðh=pÞ sþ1=2 , for the SD approximation of the VM system; where h ð¼ max K h K Þ is the mesh size and p ð¼ max K p K Þ is the spectral order. Our estimates rely on the local version with h K being the diameter of the phase-space-time element K and p K the spectral order for K. The optimal hp estimates require an exact solution in the Sobolev space H sþ1 ðXÞ: Numerical implementations, performed for examples in one space-and two velocity dimensions, are justifying the robustness of the theoretical results.

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“…However, in one and one-half geometry, relying on d'Alembert formula Schauder/Brouwer fixed point approach, is unnecessary. The fixed point approach, which was first introduced by Ukai and Okabe in [24] for the Vlasov-Poisson system, is performed for the Vlasov-Maxwell system in [21] in full details and therefore is omitted in here.…”

Section: Introductionmentioning

confidence: 99%

“…Note that for the well-posedness of the discrete solution the existence and uniqueness is due to [21], whereas the stability of the approximation scheme is justified throughout Sections 3 and 4.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov–Maxwell system

Asadzadeh

Standar

2017

Bit Numer Math

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This paper concerns a posteriori error analysis for the streamline diffusion (SD) finite element method for the one and one-half dimensional relativistic VlasovMaxwell system. The SD scheme yields a weak formulation, that corresponds to an add of extra diffusion to, e.g. the system of equations having hyperbolic nature, or convection-dominated convection diffusion problems. The a posteriori error estimates rely on dual formulations and yield error controls based on the computable residuals. The convergence estimates are derived in negative norms, where the error is split into an iteration and an approximation error and the iteration procedure is assumed to converge.

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On streamline diffusion schemes for the one and one-half dimensional relativistic Vlasov–Maxwell system

Cited by 3 publications

References 9 publications

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

Convergence of hp-Streamline Diffusion Method for Vlasov–Maxwell System

A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov–Maxwell system

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