1,975 research outputs found
A discontinuous Galerkin method for the Vlasov-Poisson system
A discontinuous Galerkin method for approximating the Vlasov-Poisson system
of equations describing the time evolution of a collisionless plasma is
proposed. The method is mass conservative and, in the case that piecewise
constant functions are used as a basis, the method preserves the positivity of
the electron distribution function and weakly enforces continuity of the
electric field through mesh interfaces and boundary conditions. The performance
of the method is investigated by computing several examples and error estimates
associated system's approximation are stated. In particular, computed results
are benchmarked against established theoretical results for linear advection
and the phenomenon of linear Landau damping for both the Maxwell and Lorentz
distributions. Moreover, two nonlinear problems are considered: nonlinear
Landau damping and a version of the two-stream instability are computed. For
the latter, fine scale details of the resulting long-time BGK-like state are
presented. Conservation laws are examined and various comparisons to theory are
made. The results obtained demonstrate that the discontinuous Galerkin method
is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86
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Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Dynamical problems and phase transitions
Issued as Financial status report, Technical reports [nos. 1-12], and Final report, Project B-06-68
A multiscale method for heterogeneous bulk-surface coupling
In this paper, we construct and analyze a multiscale (finite element) method
for parabolic problems with heterogeneous dynamic boundary conditions. As
origin, we consider a reformulation of the system in order to decouple the
discretization of bulk and surface dynamics. This allows us to combine
multiscale methods on the boundary with standard Lagrangian schemes in the
interior. We prove convergence and quantify explicit rates for low-regularity
solutions, independent of the oscillatory behavior of the heterogeneities. As a
result, coarse discretization parameters, which do not resolve the fine scales,
can be considered. The theoretical findings are justified by a number of
numerical experiments including dynamic boundary conditions with random
diffusion coefficients
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