165 research outputs found
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime
We apply the two-scale formulation approach to propose uniformly accurate
(UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic
limit regime. The nonlinear Dirac equation involves two small scales
and with in the nonrelativistic
limit regime. The small parameter causes high oscillations in time which brings
severe numerical burden for classical numerical methods. We transform our
original problem as a two-scale formulation and present a general strategy to
tackle a class of highly oscillatory problems involving the two small scales
and . Suitable initial data for the two-scale
formulation is derived to bound the time derivatives of the augmented solution.
Numerical schemes with uniform (with respect to )
spectral accuracy in space and uniform first order or second order accuracy in
time are proposed. Numerical experiments are done to confirm the UA property.Comment: 22 pages, 6 figures. To appear on Communications in Mathematical
Science
Asymptotic preserving schemes for highly oscillatory kinetic equation
This work is devoted to the numerical simulation of a Vlasov-Poisson model
describing a charged particle beam under the action of a rapidly oscillating
external electric field. We construct an Asymptotic Preserving numerical scheme
for this kinetic equation in the highly oscillatory limit. This scheme enables
to simulate the problem without using any time step refinement technique.
Moreover, since our numerical method is not based on the derivation of the
simulation of asymptotic models, it works in the regime where the solution does
not oscillate rapidly, and in the highly oscillatory regime as well. Our method
is based on a "double-scale" reformulation of the initial equation, with the
introduction of an additional periodic variable
Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field
International audienceThis paper deals with the numerical resolution of the Vlasov-Poissonsystem with a strong external magnetic field by Particle-In-Cell(PIC) methods. In this regime, classical PIC methods are subject tostability constraints on the time and space steps related to the smallLarmor radius and plasma frequency. Here, we propose anasymptotic-preserving PIC scheme which is not subjected to theselimitations. Our approach is based on first and higher order semi-implicit numericalschemes already validated on dissipative systems. Additionally, when the magnitude of the external magneticfield becomes large, this method provides a consistent PICdiscretization of the guiding-center equation, that is, incompressibleEuler equation in vorticity form. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts
Nonlinear Evolution Equations: Analysis and Numerics
The qualitative theory of nonlinear evolution equations is an
important tool for studying the dynamical behavior of systems in
science and technology. A thorough understanding of the complex
behavior of such systems requires detailed analytical and numerical
investigations of the underlying partial differential equations
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