We investigate the relative performance of a second-order Adams–Bashforth scheme and second-order and
fourth-order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a
spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments
explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of
uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a
CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability.
The number of steps, together with the order of the scheme, affects not only the runtime but also the
accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of
spatial resolution and choice of time integration on performance and provide general guidelines on how best
to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role
played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident,
especially for uniform meshes, compared with what has been typically considered when studying this type
of problem.