On the Use of Adaptive Meshes to Counter Overshoot in Solutions of Discretised Nonlinear Stochastic Differential Equations

Abstract We consider two classes of nonlinear stochastic differential equation with a.s. positive solutions. In the first case the drift coefficient is strongly zero-reverting, and dominates the diffusion, whereas in the second the diffusion is highly variable and dominates the drift. In each case, the tendency to overshoot zero prevents a uniform Euler discretisation from preserving positivity in solutions. To address this, we construct adaptive meshes allowing the generation of positive trajectories with arbitrarily high probability. For completeness, we generalise the analysis to finite-dimensional systems of stochastic differential equations, investigating the effect of a uniform Euler discretisation on the positivity of systems with coefficients satisfying linear bounds, and introducing an adaptive mesh to counter overshoot when those bounds are violated

Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations

Hybrid stochastic kinetic description of two-dimensional traffic dynamics

In this work we present a two-dimensional kinetic traffic model which takes into account speed changes both when vehicles interact along the road lanes and when they change lane. Assuming that lane changes are less frequent than interactions along the same lane and considering that their mathematical description can be done up to some uncertainty in the model parameters, we derive a hybrid stochastic Fokker-Planck-Boltzmann equation in the quasi-invariant interaction limit. By means of suitable numerical methods, precisely structure preserving and direct Monte Carlo schemes, we use this equation to compute theoretical speed-density diagrams of traffic both along and across the lanes, including estimates of the data dispersion, and validate them against real data

A positivity-preserving scheme for fluctuating hydrodynamics

A finite-difference hybrid numerical method for the solution of the isothermal fluctuating hydrodynamic equations is proposed. The primary focus is to ensure the positivity-preserving property of the numerical scheme, which is critical for its functionality and reliability especially when simulating fluctuating vapour systems. Both cases of single- and two-phase flows are considered by exploiting the van der Waals' square-gradient approximation to model the fluid (often referred to as “diffuse-interface” model). The accuracy and robustness of the proposed scheme is verified against several benchmark theoretical predictions for the statistical properties of density, velocity fluctuations and liquid-vapour interface, including the static structure factor of the density field and the spectrum of the capillary waves excited by thermal fluctuations at interface. Finally, the hybrid scheme is applied to the challenging bubble nucleation process, and is shown to capture the salient features of the phenomenon, namely nucleation rate and subsequent bubble-growth dynamics