A stable finite-difference scheme for growth and diffusion on a map

TL;DR

This paper introduces a stable, semi-implicit finite-difference scheme based on Godunov splitting for simulating growth and diffusion processes, exemplified by modeling late Pleistocene human dispersal on geographical maps.

Contribution

It presents a novel, stable numerical method suitable for modeling population dynamics on complex maps, with efficient performance and applicability to real-world historical data.

Findings

Method is stable and approximately second order accurate.

Efficient in terms of memory and computational performance.

Successfully applied to model late Pleistocene human dispersal.

Abstract

We describe a general Godunov type splitting for numerical simulations of the Fisher/Kolmogorov-Petrovski-Piskunov growth and diffusion equation in two spatial dimensions. In particular, the method is appropriate for modeling population growth and dispersal on a terrestrial map. The procedure is semi-implicit, hence quite stable, and approximately second order accurate, excluding boundary condition complications. It also has low memory requirements and shows good performance. We illustrate an application of this solver: global human dispersal in the late Pleistocene, modeled via growth and diffusion over geographical maps of paleovegetation and paleoclimate.