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Journal Articles Numerische Mathematik Year : 2007

## Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system

Martin Campos Pinto
Michel Mehrenberger

#### Abstract

An adaptive semi-Lagrangian scheme for solving the Cauchy problem associated to the periodic 1+1-dimensional Vlasov-Poisson system in the two- dimensional phase space is proposed and analyzed. A key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one, based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow. The accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step, in the L infinity metric. The numerical solutions are proved to converge in L infinity towards the exact ones as ε and Δt tend to zero provided the initial data is Lipschitz and has a finite total curvature, or in other words, that it belongs to ${W^{1,\infty} \cap W^{2,1}}$ . The rate of convergence is ${\mathcal{O}({\Delta}t^2 + \varepsilon/{\Delta}t)}$ , which should be compared to the results of Besse who recently established in (SIAM J Numer Anal 42(1):350–382, 2004) similar rates for a uniform semi-Lagrangian scheme, but requiring that the initial data are in ${{\mathcal C}^2}$ . Several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations.

#### Domains

Mathematics [math] Numerical Analysis [math.NA]

### Dates and versions

hal-00320364 , version 1 (10-09-2008)

### Identifiers

• HAL Id : hal-00320364 , version 1
• DOI :

### Cite

Martin Campos Pinto, Michel Mehrenberger. Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system. Numerische Mathematik, 2007, 108 (3), pp.407-444. ⟨10.1007/s00211-007-0120-z⟩. ⟨hal-00320364⟩

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