Abstract: We 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.

Abstract: Random Boolean networks (RBNs) are frequently employed for modelling complex systems driven by information processing, e.g. for gene regulatory networks (GRNs). Here we propose a hierarchical adaptive RBN (HARBN) as a system consisting of distinct adaptive RBNs – subnetworks – connected by a set of permanent interlinks. Information measures and internal subnetworks topology of HARBN coevolve and reach steady-states that are specific for a given network structure. We investigate mean node information, mean edge information as well as a mean node degree as functions of model parameters and demonstrate HARBNs ability to describe complex hierarchical systems.

Abstract: This study presents numerical algorithms for solving a class of equations that partly consists of derivatives of the unknown state at previous certain times, as well as an integro-differential term containing a weakly singular kernel. These equations are types of integro-differential equation of the second kind and were originally obtained from an aeroelasticity problem. One of the main contributions of this study is to propose numerical algorithms that do not involve transforming the original equation into the corresponding Volterra equation, but still enable the numerical solution of the original equation to be determined. The feasibility of the proposed numerical algorithm is demonstrated by applying examples in measuring the maximum errors with exact solutions at every computed nodes and calculating the corresponding numerical rates of convergence thereafter.

Abstract: Batteries are highly complex electrochemical systems, with performance and safety governed by coupled nonlinear electrochemical-electrical-thermal-mechanical processes over a range of spatiotemporal scales. We describe a new, open source computational environment for battery simulation known as VIBE - the Virtual Integrated Battery Environment. VIBE includes homogenized and pseudo-2D electrochemistry models such as those by Newman-Tiedemann-Gu (NTG) and Doyle-Fuller-Newman (DFN, a.k.a. DualFoil) as well as a new advanced capability known as AMPERES (Advanced MultiPhysics for Electrochemical and Renewable Energy Storage). AMPERES provides a 3D model for electrochemistry and full coupling with 3D electrical and thermal models on the same grid. VIBE/AMPERES has been used to create three-dimensional battery cell and pack models that explicitly simulate all the battery components (current collectors, electrodes, and separator). The models are used to predict battery performance under normal operations and to study thermal and mechanical response under adverse conditions.

Abstract: A magnetosphere is a region of space filled with plasma around a magnetized object, shielding it from solar wind particles. The shape of a magnetosphere is determined by the microscopic interaction phenomena between the solar wind and the dipolar magnetic field of the object. To correctly describe these interactions, we need to model phenomena occurring over a large range of time and spatial scales. In fact, magnetosphere comprises regions with different particle densities, temperatures and magnetic field where the characteristic time scales (plasma period, electron and ion gyro period) and spatial scales (Debye length, ion and electron skin depth) vary considerably. We simulate the formation of a magnetosphere with an implicit Particle-in-Cell code, called iPIC3D. We used a dipole model to represent the magnetic field of the object, where an interplanetary magnetic field is convected by the solar wind. We carried out global Particle-in-Cell simulations that consist of a complete Magnetosphere system, including magnetopause, magnetosheath and magnetotail. In this paper we describe the new algorithms implemented in iPIC3D to address the problem of modelling multi-scale phenomena in magnetosphere. In particular, we present a new adaptive sub-cycling technique to correctly describe the motion of particles that are close to the magnetic dipole. We also implemented new boundary conditions to model the inflow and outflow of solar wind in the simulation box. Finally, we discuss about the application of these new methods for modelling planetary magnetospheres.