In this paper, a global positive-definite Lyapunov function is constructed analytically for the set of non-linear partial differential algebraic equations that describe a physics-based model of a supercapacitor. The model uses a set of conservation and diffusion equations to describe ion transport in the electrodes and separator. Global stability of the model equations can be shown through Lyapunov analysis. Local asymptotic stability of the equations can also be shown when the concentration is constrained to some set. This analysis can be used to study the stability of more complex electrochemical devices, such as lithium ion batteries, which are based on similar physical relationships. It can also be used as the basis for an advanced PDE boundary charging control system whose performance will be independent of model discretisation.