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We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting…
This work discusses the numerical aspects of representing the diffusional (condensational) growth in particulate systems such as atmospheric clouds. It focuses on the Eulerian modeling approach, in which the evolution of the particle size…
In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part…
We propose a novel numerical approach for nonlocal diffusion equations [8] with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by L\`{e}vy…
A mathematical model to describe the growth of an arbitrarily large number of nanocrystals from solution is presented. First, the model for a single particle is developed. By non-dimensionalising the system we are able to determine the…
We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the…
The phenomenon of burst nucleation in solution, in which a period of apparent chemical inactivity is followed by a sudden and explosive growth of nucleated particles from a solute species, has been given a widely accepted qualitative…
Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities…
We survey our research on modeling the mechanisms of control of uniformity in growth of nanosize and colloid particles. The former are produced as nanocrystals, by burst-nucleation from solution. The latter, colloid-size particles, are…
The diffusion model has emerged as a powerful tool for generating atomic structures for materials science. This work calls attention to the deficiency of current particle-based diffusion models, which represent atoms as a point cloud, in…
We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical…
The newly developed "void expansion method" allows for an efficient generation of porous packings of spherical particles over a wide range of volume fractions using the discrete element method. Particles are randomly placed under addition…
We report a new approach to achieve growth of highly crystalline nickel nanoparticles over an extended range of sizes (up to 100 nm in diameter) and time scales (up to several hours) by diffusional transport of constituent atoms. The…
We review the numerical techniques for ideal and non-ideal magneto-hydrodynamics (MHD) used in the context of star formation simulations. We outline the specific challenges offered by modeling star forming environments, which are dominated…
In this paper we discuss numerical methods and algorithms for the solution of NLTE stellar atmosphere problems involving expanding atmospheres, e.g., found in novae, supernovae and stellar winds. We show how a scheme of nested iterations…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
Surface growth, i.e., the addition or removal of mass from the boundary of a solid body, occurs in a wide range of processes, including the growth of biological tissues, solidification and melting, and additive manufacturing. To understand…