Very simple linearisations for the solution to the Riemann problem for the time-dependent and for the steady supersonic Euler equations are presented. When used locally in conjunction with Godunov-type methods, computing savings by a factor of about four, relative to the use of exact Riemann solvers, can be achieved. For severe flow regimes however, the linearisation looses accuracy and robustness. We then propose the use of a Riemann-solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearised Riemann solver. Also, reliable and simple switching criteria are presented. Numerical results for one, two and three-dimensional test problems suggest that the resulting numerical methods are competitive for practical applications, in terms of robustness, accuracy and computational efficiency.Cranfield Institute of Technology