136 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
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
Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps
Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.</p
Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations
International audienceWe present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions
Hierarchical Micro-Macro Acceleration for Moment Models of Kinetic Equations
Fluid dynamical simulations are often performed using cheap macroscopic
models like the Euler equations. For rarefied gases under near-equilibrium
conditions, however, macroscopic models are not sufficiently accurate and a
simulation using more accurate microscopic models is often expensive. In this
paper, we introduce a hierarchical micro-macro acceleration based on moment
models that combines the speed of macroscopic models and the accuracy of
microscopic models. The hierarchical micro-macro acceleration is based on a
flexible four step procedure including a micro step, restriction step, macro
step, and matching step. We derive several new micro-macro methods from that
and compare to existing methods. In 1D and 2D test cases, the new methods
achieve high accuracy and a large speedup
Coarse Molecular Dynamics of a Peptide Fragment: Free Energy, Kinetics, and Long-Time Dynamics Computations
We present a ``coarse molecular dynamics'' approach and apply it to studying
the kinetics and thermodynamics of a peptide fragment dissolved in water. Short
bursts of appropriately initialized simulations are used to infer the
deterministic and stochastic components of the peptide motion parametrized by
an appropriate set of coarse variables. Techniques from traditional numerical
analysis (Newton-Raphson, coarse projective integration) are thus enabled;
these techniques help analyze important features of the free-energy landscape
(coarse transition states, eigenvalues and eigenvectors, transition rates,
etc.). Reverse integration of (irreversible) expected coarse variables backward
in time can assist escape from free energy minima and trace low-dimensional
free energy surfaces. To illustrate the ``coarse molecular dynamics'' approach,
we combine multiple short (0.5-ps) replica simulations to map the free energy
surface of the ``alanine dipeptide'' in water, and to determine the ~ 1/(1000
ps) rate of interconversion between the two stable configurational basins
corresponding to the alpha-helical and extended minima.Comment: The article has been submitted to "The Journal of Chemical Physics.
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