The time-dependent Euler equations of gas dynamics are a set of nonlinear 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, linearized 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 weighted average flux numerical method to solve the time-dependent Euler equations in one and two space dimensions with general initial data. For flows with shock waves of moderate strength the computed results are very accurate. For severe flow régimes we advocate the use of the present linearized 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-dimensional and two-dimensional test problems show that the linearized 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 linearized Riemann solver. The relevance of the present methods concerns the numerical solution of multi-dimensional problems accurately and economically.
