The time-dependent Euler equations of Gas Dynamics are a set of non-linear hyperbolic conservation laws that admit discontinuous solutions (e.g. shocks). In this paper we are concerned with Riemann-problem based numerical methods for solving the general initial-value problem for these equations. We present an approximate, linearised Riemann solver for the time-dependent Euler equations. The solution is direct and involves few and simple arithmetic operations. The Riemann solver is then used, locally, in conjunction with the WAF numerical method to solve the time-dependent Euler equations in one and two space dimensions with general initial data. For flows with shocks waves of moderate strength the computed results are very accurate. For severe flow regimes we advocate the use of the present linearised Riemann solver in combination with the exact Riemann solver in an adaptive fashion. Numerical experiments demonstrate that such an approach can be very successful. One and two-dimensional test problems show that the linearised Riemann solver is used in over 99% of the flow field producing net computing savings by a factor of about 2. A reliable and simple switching criterion is also presented. Results show that the adaptive approach effectively provides the resolution and robustness of the exact Riemann solver at the computing cost of the simple linearised Riemann solver. The relevance of the present methods concerns the numerical solution of multi-dimensional problems accurately and economically.Cranfield Institute of Technology