120 research outputs found
A convergent method for linear half-space kinetic equations
We give a unified proof for the well-posedness of a class of linear
half-space equations with general incoming data and construct a Galerkin method
to numerically resolve this type of equations in a systematic way. Our main
strategy in both analysis and numerics includes three steps: adding damping
terms to the original half-space equation, using an inf-sup argument and
even-odd decomposition to establish the well-posedness of the damped equation,
and then recovering solutions to the original half-space equation. The proposed
numerical methods for the damped equation is shown to be quasi-optimal and the
numerical error of approximations to the original equation is controlled by
that of the damped equation. This efficient solution to the half-space problem
is useful for kinetic-fluid coupling simulations
First-order aggregation models with alignment
We include alignment interactions in a well-studied first-order
attractive-repulsive macroscopic model for aggregation. The distinctive feature
of the extended model is that the equation that specifies the velocity in terms
of the population density, becomes {\em implicit}, and can have non-unique
solutions. We investigate the well-posedness of the model and show rigorously
how it can be obtained as a macroscopic limit of a second-order kinetic
equation. We work within the space of probability measures with compact support
and use mass transportation ideas and the characteristic method as essential
tools in the analysis. A discretization procedure that parallels the analysis
is formulated and implemented numerically in one and two dimensions
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