Numerical Approximation and Stability Analysis of Reaction–Diffusion Equations on Distance-Regular Graphs

Abstract

We study the numerical approximation of a scalar reaction–diffusion equation posed on the vertex set of a distance-regular graph, where diffusion is modelled by the graph Laplacian. For the explicit Euler scheme and the IMEX Euler scheme we ask how much of the numerical behaviour can be predicted from the intersection array of the graph alone. We prove that the implicit propagator belongs to the Bose–Mesner algebra, has positive entries that depend only on the distance between vertices, and coincides with a polynomial of degree at most the diameter in the Laplacian, so that it can be applied exactly without matrix factorisation. We derive sharp step-size limits for the explicit scheme in the Euclidean and maximum norms and show that they coincide precisely for bipartite graphs. We also obtain the exact linearised stability threshold at a homogeneous equilibrium, a discrete homogenisation estimate governed by the algebraic connectivity, and a first-order error bound. Experiments on ten distance-regular graphs, including Cayley graphs on dihedral and elementary abelian groups, confirm the predicted thresholds with a relative gap of order