468 research outputs found
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
Pointwise Green's function bounds and stability of relaxation shocks
We establish sharp pointwise Green's function bounds and consequent
linearized and nonlinear stability for smooth traveling front solutions, or
relaxation shocks, of general hyperbolic relaxation systems of dissipative
type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability,
i.e., stable point spectrum of the linearized operator about the wave, and
hyperbolic stability of the corresponding ideal shock of the associated
equilibrium system. This yields, in particular, nonlinear stability of weak
relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The
techniques of this paper should have further application in the closely related
case of traveling waves of systems with partial viscosity, for example in
compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp.
energy estimates of Section 7, corrected bad forward references, expanded
Remark 1.17, end of introductio
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