Research
Random Blob Methods for Diffusion
Katy Craig and Claire Murphy · arXiv preprint, 2026
With my Ph.D. advisor Katy Craig, I recently finished a preprint on “Random blob methods for diffusion.” While many recent works have shown that blob methods provide an attractive meshfree method for a range of linear and nonlinear diffusion equations [CJT], a key weakness of this approach is the $O(N^2)$ complexity in the number of particles $N$. In our work, Katy and I investigated two potential stochastic alternatives to reduce the complexity: the well-known random batch method [JLL] and a novel alternative, which we call the random multirate method. Interestingly, for the linear and nonlinear diffusion equations we consider (particularly in the slow diffusion regime), the random multirate method strongly outperforms alternative approaches, offering significant computational savings while preserving accuracy of the underlying dynamics and long time behavior. Code for reproducing our experiments is available in our Github repository.
Animations
In the following animations, we use the random multirate method to simulate particle solutions to several PDEs of interest. See our preprint for more details of these simulations, and solutions to many other diffusive PDEs.
$d = 1$ height-constrained transport with ramp potential $V(x) = \min(x, 0)$. Formally, the solution to the PDE moves according to the transport equation, up to the restriction that the density of the solution can never exceed height one. Our initial condition is two connected components. Up to time $t_{\text{crit}} := 0.5$, the left component moves to the right with speed one. After time $t_{\text{crit}}$, we can think of this left component "pushing" the right component farther to the right. The fact that the density can continue to evolve on regions where the velocity is zero is a key feature of our version of height-constrained transport.
Sandpile dynamics with critical height $r_c = 0.1$. Heuristically, solutions of this PDE behave as follows: if the density exceeds the critical height threshold $r_c > 0$ on a given region, it diffuses on that region, but there is no diffusion when the density is beneath the critical threshold.
The viscosity formulation of the 2D Navier-Stokes equation with a "double bump" initial condition. While [Nor] relied on regridding to simulate this PDE, we recover the same behavior without it. The dots shown correspond to a select subset of particles in our simulation.
Cited Literature
[CJT] Craig, Katy, Matt Jacobs, and Olga Turanova. Nonlocal Approximation of Slow and Fast Diffusion. Journal of Differential Equations, Volume 42 (2025).
[JLL] Jin, Shi, Lei Li, and Jian-Guo Liu. Random Batch Methods (RBM) for interacting particle systems. Journal of Computational Physics, Volume 400 (2019).
[Nor] Nordmark, Henrik O. Deterministic High Order Vortex Methods for the 2D Navier–Stokes Equation with Rezoning. Journal of Computational Physics, Volume 129 (1996).