Linearly implicit schemes for convection-diffusion equations

We present a family of schemes for the approximation of one dimensional convection-diffusion equations. It is based on a linearization technique that allows to treat explicitly the hyperbolic term and linearly implicitly the parabolic one. This avoids the parabolic stability constraint of explicit schemes, and does not require any non-linear solver for the implicit problem. We present several numerical simulations to show the effectiveness of the proposed schemes and to investigate their stability, convergence and accuracy. In particular, since the proposed schemes provide to be accurate for both smooth and non-smooth solutions, they turn out to be attractive for adaptivity