arXiv subjects

Troy Tsubota

Publications and source records attributed to Troy Tsubota.

2 recordsLinked to original sources

arXiv2026

Accurate simulation of pulled and pushed fronts in the nonautonomous Fisher-KPP equation

We introduce a novel numerical method for direct simulation of front propagation in the Fisher-KPP equation with a time-dependent parameter on an infinite domain. The method computes a time-dependent boundary condition that accurately captures the leading-edge dynamics by coupling the nonlinear simulation region to a linear approximation region in which the dynamics can be solved exactly via the Green's function of the linearized equation. This approach enables precise front velocity measurements on relatively small computational domains for a variety of nonautonomous regimes and initial conditions for which existing numerical methods break down. We apply the method to pulled and pushed fronts in the Fisher-KPP equation with quadratic and quadratic-cubic nonlinearities, finding that it improves the accuracy of the simulated front velocity even for constant parameters and a fixed domain size. For pulled fronts with a diffusion coefficient that increases algebraically in time, our results reveal a deviation from the natural asymptotic velocity predicted by linear theory, whose explanation requires nonlinear theory. For pushed fronts with constant parameters, the method reproduces the exponential convergence to the theoretical asymptotic front speed and profile with improved precision. For a slowly time-varying linear growth parameter, we find that the pushed front velocity follows the changing parameter adiabatically if the asymptotic pushed velocity remains faster than the natural asymptotic pulled velocity. As the growth parameter moves toward the pushed--pulled transition point, the competition between the pushed and pulled fronts can result in both delayed and even premature onset of the pushed--pulled transition, depending on the form of parameter growth. The numerical method presented here proves to be an effective tool for analyzing front propagation in nonautonomous systems.

physics.flu-dyn↗

arXiv2023

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation (RGLE) with periodic boundary conditions on monotonically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e. phase slips that fail to develop. Then, we study how fronts are influenced by a time-dependent domain. We derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural ``asymptotic'' velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time-dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern-spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution -- an effect present when the order parameter is conserved -- increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains.

nlin.PS↗