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.

💡 Research Summary

The paper introduces a novel numerical scheme for directly simulating front propagation in the one‑dimensional Fisher‑KPP equation with time‑dependent parameters on an infinite spatial domain. The authors recognize that the leading edge of a front is governed by linear dynamics, and they exploit this fact by splitting the computational domain into two overlapping regions: a nonlinear (NL) region where the full equation is solved, and a linear approximation (LA) region extending to infinity where the equation is linearized about the unstable state u = 0. In the LA region the solution can be expressed analytically using the Green’s function of the linearized operator. By convolving the Green’s function with the boundary value supplied by the NL region, the method yields an exact expression for the solution at the outer boundary of the NL region. This value is then imposed as a time‑dependent Dirichlet condition for the next NL‑region time step. The approach eliminates spurious reflections and captures the exponential tail u ∼ e^{−λx} of pulled fronts without requiring a priori knowledge of the steepness λ.

Implementation details include second‑order central finite differences in space, an IMEX Euler time integrator (linear terms treated implicitly, nonlinear terms explicitly), and a buffer of width δ to regularize the convolution. The authors assume a proportionality between the moving‑frame speed c(t) and the diffusion coefficient d(t) (c = γ d) to obtain closed‑form Green’s functions; γ can be tuned to keep the leading edge small throughout the simulation. The method is computationally efficient because the convolution integrals can be evaluated analytically or with inexpensive quadrature, allowing accurate simulations on modest domain sizes (e.g., x∈