In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semialgebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets.

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The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann - Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D $n^+-n-n^+$ silicon diodes with $400nm$ and $50nm$ channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).

In this paper, we apply discontinuous Galerkin (DG) methods to solve two model equations: Krause's consensus models and pressureless Euler equations. These two models are used to describe the collisions of particles, and the distributions can be identified as density functions. If the particles are placed at a single point, then the density function turns out to be a $\delta$-function and is difficult to be well approximated numerically. In this paper, we use DG method to approximate such a singularity and demonstrate the good performance of the scheme. Since the density functions are always positive, we apply a positivity-preserving limiter to them. Moreover, for pressureless Euler equations, the velocity satisfies the maximum principle. We also construct special limiters to fulfill this requirement.

Abstract In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $\ell^1$ contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $ (\Delta t)^{\alpha} = O(\Delta x)$, where $\alpha \in (0,1]$ is the fractional exponent in the derivative. When $\alpha$ small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $\ell^1$ contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.