An unsteady one-dimensional model of solid propellant combustion, based on a low-Mach assumption, is presented and semi-discretised in space via a finite volume scheme. The mathematical nature of this system is shown to be differential-algebraic of index one. A high-fidelity numerical strategy with stiffly accurate singly diagonally implicit Runge-Kutta methods is proposed, and time adaptation is made possible using embedded schemes. High-order is shown to be reached, while handling the constraints properly, both at the interface and for the mass conservation in the gaseous flow field. Three challenging test-cases are thoroughly investigated: ignition transients, growth of combustion instabilities through a Hopf bifurcation leading to a limit cycle periodic solution and the unsteady response of the system when detailed gas-phase kinetics are included in the model. The method exhibits high efficiency for all cases in terms of both computational time and accuracy compared to first and second-order schemes traditionally used in the combustion literature, where the time step adaptation is CFL-or variation-based.