Numerical computation of an Evans function for travelling waves

•A geometrically inspired technique for computing Evans functions.•Evans functions are computed for scalar and systems of PDEs.•Numerically examine stability of travelling waves in a chemotaxis model.•Includes a new proof of stability of travelling waves in the F-KPP equation.

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller–Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c≥2δ in the F-KPP equation.