VisualPDE: Rapid Interactive Simulations of Partial Differential Equations

by Benjamin J. Walker, Adam K. Townsend, Alexander K. Chudasama & Andrew L. Krause

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Mathematical biology and other areas of science are employing increasingly complex models that take the form of partial differential equations. Such models can exhibit a rich set of behaviours, including those that defy intuition, such as diffusion-driven pattern formation. In this paper we present VisualPDE, a web-based tool enabling  real-time interactive exploration of such models. We feel that such interactive 'play' is an incredibly important and under-utilized way to develop intuition and build deep understanding of these models.

The paper opens by saying that a reader should simply go and play on the website, VisualPDE.com, themselves. It then outlines our rationale for developing this tool, some of the technical and design issues we faced, as well as some of the examples and use-cases we have already explored. The structure of this paper supplements the 'living' website with things we think a reader might find interesting, particularly around the wider context and technical aspects of designing the website. We hope it helps the wider community deepen our understanding of PDE models, and rethink how we teach and do research using mathematics more generally.

Image caption: This is an interactive simulation on the website which can also be viewed at https://visualpde.com/sim/?preset=BMB . It uses spatial heterogeneity to force a reaction-diffusion system to exhibit both a complex prepattern and emergent spot-like patterns in different parts of the domain.