We propose a new fractional threshold-dynamics algorithm for subdiffusive interface motion and apply it to multiphase grain-growth simulations. The method combines the L1 discretization of the Caputo time derivative with a Helmholtz-resolvent thresholding step. At each time level, the L1 convolution produces a convex history average of the previously thresholded states; this history-dependent field is then propagated by a Helmholtz diffusion solve and projected back to pure phases by thresholding. For $\alpha=1$, the method reduces to a Helmholtz-resolvent analogue of the Merriman-Bence-Osher scheme, whereas for $0<\alpha<1$ it introduces a subdiffusive temporal memory. We establish a variational characterization, maximum and comparison principles, and stability for the continuous Helmholtz diffusion step. We also derive a memory-tail estimate and a finite-dimensional pinning criterion under an explicit resolvent-contraction hypothesis. The equal-tension multiphase scheme is implemented on periodic grids. The numerical experiments investigate the joint influence of the L1 history and the smoothing length. The fixed-step runs show faster coarsening for smaller fractional orders in the tested regime, while a comparison at fixed smoothing length examines the influence of the L1 history on successive updates.
Subdiffusion occurs when long trapping or residence times of particles slow down spatial transport and produce a mean squared displacement proportional to $t^\alpha$, with~$0<\alpha<1$. We develop a general renewal--jump framework that combines the internal trapping dynamics with the reinjection of particles with spati...
Benoît Perthame Ljll, Musclees, M. Tang· 0 citations
In this paper, a lattice Boltzmann method with an adaptive relaxation parameter and a Lax--Friedrichs-type equilibrium is proposed for hyperbolic systems with source terms. The equilibrium distribution recovers the conservative variables and physical fluxes while incorporating dissipation determined by characteristic-s...
Yi-Fan Zhang, Ping Lin, Jin Zhao et al.· arXiv.org· 0 citations
As random displacements of many biological entities such as animals and cells often display statistical characteristics that depart from diffusion, over the years various mathematical representations to describe the so-called anomalous movement statistics have been developed. In the subdiffusive regime, that is when th...
Haniyyah Ul Irshad, L. Giuggioli, H. Fahad et al.· Journal of Physics A: Mathem...· 0 citations
We study a coupled Stokes-reaction-diffusion-advection system in a three dimensional domain perforated by a large number of evolving spherical inclusions with radii of order $\varepsilon^\alpha$ for $\alpha\in(1,3]$. Their centers are separated on the order of $\varepsilon$ but are not assumed to form a periodic lattic...
In this paper, we extend numerical schemes based on parameterized Young measures and linear programming to two space dimensions and apply them to the Kelvin-Helmholtz instability governed by the two-dimensional Euler equations of gas dynamics. We construct first-, second-, third-, fifth-, seventh-, and ninth-order Youn...
Phase-field simulations of solid-state dewetting based on polynomial potentials may exhibit spurious bulk-diffusion coarsening, inconsistent with the surface-diffusion-dominated physics. To address this issue, we formulate a degenerate Cahn--Hilliard model with the logarithmic Flory--Huggins potential and dynamic conta...
