VisualPDE: rapid interactive simulations of partial differential equations

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

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)


Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.
Original languageEnglish
Article number113
Number of pages24
JournalBulletin of Mathematical Biology
Publication statusPublished - 12 Oct 2023

Bibliographical note

Acknowledgements The ideas for this project originated in a Durham Centre for Academic Development collaborative innovation grant titled Accessible interactive visualisations in mathematical biology, which supported AKC in the initial version of the interactive PDE solver, based on the Gray–Scott reaction– diffusion simulator by pmneila (2012). BJW is supported by the Royal Commission for the Exhibition of

Data Availability There is no data present in this paper. All code associated with the project can be found on GitHub (Walker et al. 2023).


  • physics.ed-ph
  • nlin.PS
  • physics.comp-ph


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  • VisualPDE

    Walker, B., Krause, A. L. & Townsend, A.

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