Towards a unified linear kinetic transport model with the trace ion module for EIRENE

Linear kinetic Monte Carlo particle transport models are frequently employed in fusion plasma simulations to quantify atomic and surface effects on the main plasma flow dynamics. Separate codes are used for transport of neutral particles (incl. radiation) and charged particles (trace impurity ions). Integration of both modules into main plasma fluid solvers provides then self consistent solutions, in principle. The required interfaces are far from trivial, because rapid atomic processes in particular in the edge region of fusion plasmas require either smoothing and resampling, or frequent transfer of particles from one into the other Monte Carlo code. We propose a different scheme here, in which despite the inherently different mathematical form of kinetic equations for ions and neutrals (e.g. Fokker-Planck vs. Boltzmann collision integrals) both types of particle orbits can be integrated into one single code. We show that the approximations and shortcomings of this "single sourcing" concept (e.g., restriction to explicit ion drift orbit integration) can be fully tolerable in a wide range of typical fusion edge plasma conditions, and be overcompensated by the code-system simplicity, as well as by inherently ensured consistency in geometry (one single numerical grid only) and (the common) atomic and surface process modulesComment: 15 pages, 7 figure

Linear Boltzmann dynamics in a strip with large reflective obstacles: stationary state and residence time

The presence of obstacles modify the way in which particles diffuse. In cells, for instance, it is observed that, due to the presence of macromolecules playing the role of obstacles, the mean square displacement ofbiomolecules scales as a power law with exponent smaller than one. On the other hand, different situations in grain and pedestrian dynamics in which the presence of an obstacle accelerate the dynamics are known. We focus on the time, called residence time, needed by particles to cross a strip assuming that the dynamics inside the strip follows the linear Boltzmann dynamics. We find that the residence time is not monotonic with the sizeand the location of the obstacles, since the obstacle can force those particles that eventually cross the strip to spend a smaller time in the strip itself. We focus on the case of a rectangular strip with two open sides and two reflective sides and we consider reflective obstaclea into the strip

Palindromic 3-stage splitting integrators, a roadmap

The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table

Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation

In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the author's knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D

Spatially hybrid computations for streamer discharges with generic features of pulled fronts: I. Planar fronts

Streamers are the first stage of sparks and lightning; they grow due to a strongly enhanced electric field at their tips; this field is created by a thin curved space charge layer. These multiple scales are already challenging when the electrons are approximated by densities. However, electron density fluctuations in the leading edge of the front and non-thermal stretched tails of the electron energy distribution (as a cause of X-ray emissions) require a particle model to follow the electron motion. As super-particle methods create wrong statistics and numerical artifacts, modeling the individual electron dynamics in streamers is limited to early stages where the total electron number still is limited. The method of choice is a hybrid computation in space where individual electrons are followed in the region of high electric field and low density while the bulk of the electrons is approximated by densities (or fluids). We here develop the hybrid coupling for planar fronts. First, to obtain a consistent flux at the interface between particle and fluid model in the hybrid computation, the widely used classical fluid model is replaced by an extended fluid model. Then the coupling algorithm and the numerical implementation of the spatially hybrid model are presented in detail, in particular, the position of the model interface and the construction of the buffer region. The method carries generic features of pulled fronts that can be applied to similar problems like large deviations in the leading edge of population fronts etc.Comment: 33 pages, 15 figures and 2 table

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.

Numerical approximation of BSDEs using local polynomial drivers and branching processes

We propose a new numerical scheme for Backward Stochastic Differential Equations based on branching processes. We approximate an arbitrary (Lipschitz) driver by local polynomials and then use a Picard iteration scheme. Each step of the Picard iteration can be solved by using a representation in terms of branching diffusion systems, thus avoiding the need for a fine time discretization. In contrast to the previous literature on the numerical resolution of BSDEs based on branching processes, we prove the convergence of our numerical scheme without limitation on the time horizon. Numerical simulations are provided to illustrate the performance of the algorithm.Comment: 28 page

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sufficient conditions under which our probabilistic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method