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We consider a finite element method with symmetric stabilisation for thediscretisation of the transient convection--diffusion equation. For thetime-discretisation we consider either the second order backwardsdifferentiation formula or the Crank-Nicolson method. Both the convection termand the associated stabilisation are treated explicitly using an extrapolatedapproximate solution. We prove stability of the method and the $\tau^2 +h^{p+{\frac12}}$ error estimates for the $L^2$-norm under either the standardhyperbolic CFL condition, when piecewise affine ($p=1$) approximation is used,or in the case of finite element approximation of order $p \ge 1$, a stronger,so-called $4/3$-CFL, i.e. $\tau \leq C h^{4/3}$. The theory is illustrated withsome numerical examples.

Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty