13 research outputs found
Telescopic Projective Integration for Multiscale Kinetic Equations with a Specified Relaxation Profile
AbstractWe study the design of a general, fully explicit numerical method for simulating kinetic equations with an extended BGK collision model allowing for multiple relaxation times. In that case, the problem is stiff and we show that its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. They are very efficient schemes, provided there are only two clusters of eigenvalues. Telescopic projective integration methods generalize the idea of projective integration methods by constructing a hierarchy of projective levels. Here, we show how telescopic projective integration methods can be used to efficiently integrate multiple relaxation time BGK models. We show that the number of projective levels only depends on the number of clusters and the size of the outer level time step only depends on the slowest time scale present in the model. Both do not depend on the small-scale parameter. We analyze stability and illustrate with numerical results
On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations
We present a convergence proof for higher order implementations of the
projective integration method (PI) for a class of deterministic multi-scale
systems in which fast variables quickly settle on a slow manifold. The error is
shown to contain contributions associated with the length of the microsolver,
the numerical accuracy of the macrosolver and the distance from the slow
manifold caused by the combined effect of micro- and macrosolvers,
respectively. We also provide stability conditions for the PI methods under
which the fast variables will not diverge from the slow manifold. We
corroborate our results by numerical simulations.Comment: 43 pages, 7 figures; accepted for publication in the Journal of
Computational and Applied Mathematic
Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time
Projective Integration methods are explicit time integration schemes for
stiff ODEs with large spectral gaps. In this paper, we show that all existing
Projective Integration methods can be written as Runge-Kutta methods with an
extended Butcher tableau including many stages. We prove consistency and order
conditions of the Projective Integration methods using the Runge-Kutta
framework. Spatially adaptive Projective Integration methods are included via
partitioned Runge-Kutta methods. New time adaptive Projective Integration
schemes are derived via embedded Runge-Kutta methods and step size variation
while their accuracy, stability, convergence, and error estimators are
investigated analytically and numerically
Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures
We propose fully explicit projective integration and telescopic projective
integration schemes for the multispecies Boltzmann and \acf{BGK} equations. The
methods employ a sequence of small forward-Euler steps, intercalated with large
extrapolation steps. The telescopic approach repeats said extrapolations as the
basis for an even larger step. This hierarchy renders the computational
complexity of the method essentially independent of the stiffness of the
problem, which permits the efficient solution of equations in the hyperbolic
scaling with very small Knudsen numbers. We validate the schemes on a range of
scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid
instabilities, and other complex phenomena
Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations
In this article, we are interested in the asymptotic analysis of a finite
volume scheme for one dimensional linear kinetic equations, with either
Fokker-Planck or linearized BGK collision operator. Thanks to appropriate
uniform estimates, we establish that the proposed scheme is
Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the
discrete framework the hypocoercivity method proposed by [J. Dolbeault, C.
Mouhot and C. Schmeiser, Trans. Amer. Math. Soc., 367, 6 (2015)] to prove the
exponential return to equilibrium of the approximate solution. We obtain decay
rates that are bounded uniformly in the diffusive limit.
Finally, we present an efficient implementation of the proposed numerical
schemes, and perform numerous numerical simulations assessing their accuracy
and efficiency in capturing the correct asymptotic behaviors of the models.Comment: 39 pages, 10 figures, 2 table
A high-order asymptotic-preserving scheme for kinetic equations using projective integration
We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp. parabolic, limiting equation exists. The scheme first takes a few small (inner) steps
with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of
the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge–Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps
is also independent of the scaling parameter. We analyse stability and consistency, and illustrate
with numerical results.32 pages, 8 figuresstatus: publishe