High Mach number shocks are ubiquitous in interstellar turbulence. The Pencil Code is particularly well suited to the study of magnetohydrodynamics in weakly compressible turbulence and the numerical investigation of dynamos because of its high-order advection and time evolution algorithms. However, the high-order algorithms and lack of Riemann solver to follow shocks make it less well suited to handling high Mach number shocks, such as those produced by supernovae (SNe). Here, we outline methods required to enable the code to efficiently and accurately model SNe, using parameters that allow stable simulation of SN-driven turbulence, in order to construct a physically realistic galactic dynamo model. These include the resolution of shocks with artificial viscosity, thermal conductivity and mass diffusion; the correction of the mass diffusion terms and a novel generalisation of the Courant condition to include all source terms in the momentum and energy equations. We test our methods with the numerical solution of the one-dimensional (1D) Riemann shock tube, also extended to a 1D adiabatic shock with parameters and Mach number relevant to SN shock evolution, including shocks with radiative losses. We extend our test with the three-dimensional (3D) numerical simulation of individual SN remnant evolution for a range of ambient gas densities typical of the interstellar medium and compare these to the analytical solutions of Sedov–Taylor (adiabatic) and the snowplough and Cioffi et al. results incorporating cooling and heating processes. We show that our new timestep algorithm leads to linear rather than quadratic resolution dependence as the strength of the artificial viscosity varies, because of the corresponding change in the strength of interzone gradients.