A self-organizing Lagrangian particle method for adaptive-resolution advection–diffusion simulations

Abstract

We present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two-and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement. AbstractWe present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two-and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement.

A hybrid particle-mesh method for incompressible active polar viscous gels

“…For complex geometries, discretizing differential operators using simple finite difference schemes is problematic. This limitation can be addressed by using generalized finite-difference schemes, like discretization-corrected particle strength exchange (DC-PSE) operators [64,59]. In addition, the current time integration scheme is explicit and is therefore only conditionally stable.…”

Radial Basis Function Approximations of Bayesian Parameter Posterior Densities for Uncertainty Analysis

“…The fluctuations are a result of stochasticity in the interacting particle scheme [16]. This suggests that, for this example, the resolution of the approximation to ∇p(θ|D) has only a negligible influence on the resulting error of the marginal approximation.…”

“…a higher density in θ can be achieved in areas where the non-normalised density p(θ|D) exhibits large fluctuations [16].…”

Convergence of a linearly transformed particle method for aggregation equations

“…We also remark that particles methods have been combined with remeshing and adaptive mesh refinement for transport and convection-diffusion equations, see [8, 9, 69] and the references therein, which also require global transforms or mapping functions related to the distortion of the flow.…”